Birkhoff Coordinates of Integrable Hamiltonian systems in Asymptotic

Birkhoff Coordinates of Integrable Hamiltonian systems in Asymptotic Regimes Dissertation zur Erlangung der naturwissenschaftlichen Doktorwürde (Dr. sc. nat.) vorgelegt der Mathematisch-naturwissenschaftlichen Fakultät der Universität Zürich von Alberto Maspero aus Italien Promotionskomitee Prof. Dr. Thomas Kappeler , Vorsitz und Leitung Prof. Dr. Dario Bambusi , Leitung Prof. Dr. Camillo de Lellis Prof. Dr. Antonio Ponno Zürich, 2015

To my parents.

Zusammenfassung Es werden zwei integrable Hamiltonsche Systeme untersucht: das Toda-Gitter mit periodischen Randwertbedingungen und einer grossen Anzahl Partikel und die Korteweg-de Vries (KdV) Gleichung auf R. Im ersten Teil untersuchen wir das asymptotische Verhalten von Birkhoff Koordinaten (kartesische Wirkung- und Winkelvariablen) des Toda-Gitters in der Nähe des Gleichgewichts, falls die Anzahl Partikel N gegen Unendlich strebt. Wir zeigen, dass für geeignet gewählte Konstanten R, R0 > 0, die der Anzahl N entsprechende Koordinatentransformation die komplexe Kugel von Radius R/N α um den Gleichgewichtspunkt analytisch in eine Kugel mit Radius R/N α abbildet genau dann falls α ≥ 2. Dabei werden Sobolev-analytische Normen gewählt. Als Anwendung betrachten wir das Problem der Gleichverteilung der Energie für Lösungen des Toda Gitters im Sinne von FermiPasta-Ulam. Wir zeigen, dass für Anfangswerte kleiner als R/N 2 , 0 < R 1, bei denen nur der erste Fourier Koeffizient nicht null ist, die Energie für alle Zeiten in einem Wellenpacket eingeschlossen bleibt, dessen Fourierkoeffizienten exponentiell mit der Wellenzahl abfallen. Schliesslich zeigen wir, dass für Lösungen des FPU-Gitters mit den oben beschriebenen Anfangswerten die Energie für ein längeres Zeitintervall in einem derartigen Wellenpacket eingeschlossen bleibt, als zuvor bekannt war. Im zweiten Teil wird die Streuabbildung für die KdV Gleichung auf R untersucht. Es wird gezeigt, dass für Potentiale in gewichteten Sobolev Räumen ohne Eigenzustände der nichtlineare Teil der Streuabbildung 1-regularisierend ist und die sich daraus ergebende Anwendungen für die Lösungen der KdV Gleichung diskutiert.

Abstract In this thesis we investigate two examples of infinite dimensional integrable Hamiltonian systems in 1-space dimension: the Toda chain with periodic boundary conditions and large number of particles, and the Korteweg-de Vries (KdV) equation on R. In the first part of the thesis we study the Birkhoff coordinates (Cartesian action angle coordinates) of the Toda lattice with periodic boundary condition in the limit where the number N of the particles tends to infinity. We prove that the transformation introducing such coordinates maps analytically a complex ball of radius R/N α (in discrete Sobolev-analytic norms) into a ball of radius R0 /N α (with R, R0 > 0 independent of N ) if and only if α ≥ 2. Then we consider the problem of equipartition of energy in the spirit of Fermi-Pasta-Ulam. We deduce that corresponding to initial data of size R/N 2 , 0 < R 1, and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Finally we consider the original FPU model and prove that energy remains localized in a similar packet of Fourier modes for times one order of magnitude longer than those covered by previous results which is the time of formation of the packet. The proof of the theorem on Birkhoff coordinates is based on a new quantitative version of a Vey type theorem by Kuksin and Perelman which could be interesting in itself. In the second part of the thesis we study the scattering map of the KdV on R. We prove that in appropriate weighted Sobolev spaces of the form H N ∩ L2M , with integers N ≥ 2M ≥ 8 and in the case of no bound states, the scattering map is a perturbation of the Fourier transform by a regularizing operator. As an application of this result, we show that the difference of the KdV flow and the corresponding Airy flow is 1-smoothing.

Acknowledgements This thesis has been carried out in a co-tutelle program between the University of Milan and the University of Zurich. I am incredibly indebted to my advisors Professor Dario Bambusi and Professor Thomas Kappeler for their constant and generous guidance, the numerous mathematical discussions throughout the development of my thesis, and their endless patience. They have been invaluable precious mentors both in mathematics and life. A special thanks to my family. Words cannot express how grateful I am to my mother, my father and my sister for all their help and support during my whole life. I would like to thank Tetiana Savitska for her love, kindness and the infinite support she has shown during the past years. Finally, I also must acknowledge my friends and colleagues who helped me throughout my PhD giving me mathematical advice and support. They include Alberto Vezzani, Beat Schaad, Hasan Inci, Jan Molnar, Yannick Widmer, Luca Spolaor, Ivana Kosirova, Urs Wagner and Wei Xue.

Contents Introduction

6

1 Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU 1 Introduction and main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Birkhoff coordinates for the Toda lattice . . . . . . . . . . . . . . . . . . . . . 1.2 On the FPU metastable packet . . . . . . . . . . . . . . . . . . . . . . . . . . 2 A quantitative Kuksin-Perelman Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Statement of the theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Proof of the Quantitative Kuksin-Perelman Theorem . . . . . . . . . . . . . . 3 Toda lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Proof of Theorem 1.3 and Corollary 1.6. . . . . . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 FPU packet of modes: proofs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Properties of normally analytic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . B Discrete Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C Proof of Proposition 1.47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D Proof of Lemma 1.50 and Corollary 1.51 . . . . . . . . . . . . . . . . . . . . . . . . . E Proof of Proposition 1.52 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 One smoothing properties of the KdV flow on R 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2 Jost solutions . . . . . . . . . . . . . . . . . . . . . . 3 One smoothing properties of the scattering map. . . 4 Inverse scattering map . . . . . . . . . . . . . . . . . 4.1 Gelfand-Levitan-Marchenko equation . . . . 5 Proof of Corollary 2.2 and Theorem 2.3 . . . . . . . A Auxiliary results. . . . . . . . . . . . . . . . . . . . B Analytic maps in complex Banach spaces . . . . . . C Properties of the solutions of integral equation (2.93) D Proof from Section 4 . . . . . . . . . . . . . . . . . . ± D.1 Properties of Kx,σ and f±,σ . . . . . . . . . . . E Hilbert transform . . . . . . . . . . . . . . . . . . . . Bibliography

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

. . . . . . . . . . . .

12 12 13 17 19 19 24 33 33 41 44 47 51 53 54 55 67 67 71 86 96 101 107 109 111 113 118 118 123

125

5

Introduction In the last decades the problem of a rigorous analysis of the theory of infinite dimensional integrable Hamiltonian systems has been widely studied. In particular Kappeler with collaborators introduced a series of methods in order to construct rigorously action-angle variables for 1-dimensional integrable Hamiltonian PDE’s on T. The program succeeded in many cases, like KdV [KP03], defocusing and focusing NLS [GK14, KLTZ09]. In each case considered, it has been proved that there exists a real analytic symplectic diffeomorphism between two scales of Hilbert spaces which introduce action-angle coordinates. The present thesis is part of this program. Infinite dimensional integrable Hamiltonian systems in 1-space dimension come up in two setups: (i) on compact intervals (finite volume) and (ii) on infinite intervals (infinite volume). The dynamical behaviour of the systems in the two setups have many similar features, but also distinct ones, mostly due to the different manifestation of dispersion. In this thesis we analyze two systems in different setups. As an example of a system in the first setup we study the Toda chain with a large number of particles, while as an example of a system in the second setup we study the Korteweg-de Vries equation (KdV) on R. The choice of the Toda chain is motivated by the application to the FPU chain which will be discussed below, while the choice of KdV is motivated by the question if features such as the 1-smoothing property established recently for this equation in the periodic setup also hold in the infinite volume case. We describe now in more details our results. The Toda’s chain. The Toda chain is the system with Hamiltonian HT oda (p, q) =

N −1 N −1 1 X 2 X qj −qj+1 pj + e 2 j=0 j=0

(1)

and periodic boundary conditions qN = q0 , pN = p0 , which is known to be integrable. We are interested in the limit N → ∞. By standard Arnold-Liouville theory the system admits action angle coordinates. However the actual introduction of such coordinates is quite complicated and the corresponding transformation has only recently been studied analytically in a series of papers by Henrici and Kappeler [HK08b, HK08c]. In particular such authors have proved the existence of global Birkhoff coordinates, namely canonical coordinates (xk , yk ) analytic on the whole phase space, with the property that the k th action is given by (x2k +yk2 )/2. The construction of Henrici and Kappeler, however is not uniform in the size of the chain, in the sense that the map ΦN introducing Birkhoff coordinates is globally analytic for any fixed N , but it could (and actually does) develop singularities as N → +∞. Our main result is to prove some analyticity properties fulfilled by ΦN uniformly in the limit N → +∞. Precisely we consider complex balls centered at the origin and prove that ΦN maps analytically a ball of radius R/N α in discrete Sobolev-analytic norms into a ball of radius R0 /N α (with R, R0 > 0 independent of N ) if and only if α ≥ 2. To come to a precise statement we have to introduce in CN −1 × CN −1 . Consider the Toda lattice in Pa suitable topology P the subspace characterized by j qj = 0 = j pj which is invariant under the dynamics. Introduce the discrete Fourier transform F(q) = qˆ defined by N −1 1 X √ qj e2iπjk/N , qˆk = N j=0

6

k∈Z,

(2)

and consider pˆk defined analogously. Finally introduce the linear Birkhoff variables pˆk + pˆN −k − iωk (ˆ qk − qˆN −k ) pˆk − pˆN −k + iωk (ˆ qk + qˆN −k ) √ √ , Yk = , k = 1, ..., N −1 , (3) 2ωk i 2ωk k := 2 sin(kπ/N ); using such coordinates, which are symplectic, the quadratic where ωk ≡ ω N part of the Hamiltonian takes the form Xk =

H0 =

N −1 X

ω

k N

k=1

Xk2 + Yk2 . 2

(4)

For any s ≥ 0, σ ≥ 0 introduce in CN −1 × CN −1 the discrete Sobolev-analytic norm 2

k(X, Y )kP s,σ :=

N −1 1 X 2s 2σ[k]N [k]N e ω N k=1

k N

|Xk |2 + |Yk |2 2

(5)

where [k]N := min(|k|, |N − k|) . The space CN −1 × CN −1 endowed by such a norm will be denoted by P s,σ . We denote by B s,σ (R) the ball of radius R and center 0 in the topology defined by the norm k.kP s,σ . We will also denote by BRs,σ := B s,σ (R) ∩ (RN −1 × RN −1 ) the real ball of radius R. The most important result in this section is the following 0 , such that Theorem 0.1. For any s ≥ 0, σ ≥ 0 there exist strictly positive constants Rs,σ , Rs,σ for any N ≥ 2, the map ΦN is analytic as a map 0 Rs,σ Rs,σ s,σ ,→ B , (x, y) 7→ (X, Y ) ΦN : B s,σ α N Nα

if and only if α ≥ 2. The same is true for the inverse mapping Φ−1 N . In order to prove the "if" part of Theorem 0.1 we apply to the Toda lattice a Vey type theorem [Vey78] for infinite dimensional systems recently proved by Kuksin and Perelman [KP10]. Actually, we need to prove a new quantitative version of Kuksin-Perelman’s theorem, a result that we think could be interesting in itself. In order to prove the "only if" part of Theorem 0.1, we explicitly construct the first term of the Taylor expansion of ΦN through Birkhoff normal form techniques, and prove that the second differential QΦN := d2 ΦN (0, 0) at the origin diverges like N 2 . It follows that, as N → +∞, the real diffeomorphism ΦN develops a singularity at zero in the second derivative. Thus, by Cauchy estimate, the image of a ball of radius R/N α is unbounded when α < 2. We finally apply the result to the problem of equipartition of energy in the spirit of Fermi-PastaUlam. Recall that the FPU (α, β)-model is the Hamiltonian lattice with Hamiltonian function which, in suitable rescaled variables, takes the form HF P U (p, q) =

N −1 X j=0

p2j + U (qj − qj+1 ) , 2

U (x) =

x2 x3 x4 + +β . 2 6 24

(6)

We will consider the case of periodic boundary conditions: q0 = qN , p0 = pN . Let us denote by Ek the energy of the k th normal mode, and by Ek := Ek /N the specific energy in the k th mode. In 7

their celebrated numerical experiment Fermi Pasta and Ulam [FPU65] studied both the behaviour Rt of Ek (t) and of its time average hEk i(t) := 1t 0 Ek (s)ds . They observed that, corresponding to initial data with E1 (0) 6= 0 and Ek (0) = 0 ∀k 6= 1, N − 1, the quantities Ek (t) present a recurrent behaviour, while their averages hEk i(t) quickly relax to a sequence E¯k exponentially decreasing with k. This is what is known under the name of FPU packet of modes. A systematic numerical study of the evolution of the Toda compared to FPU, paying particular attention to the dependence on N of the phenomena, was performed by Benettin and Ponno [BP11] (see also [BCP13]). In particular such authors put into evidence the fact that the FPU packet seems to have an infinite lifespan in the Toda lattice. Furthermore they showed that the relevant parameter controlling the lifespan of the packet in the FPU model is the distance of FPU from the corresponding Toda lattice, which is measured by the quantity (β − 1). As a corollary of Theorem 0.1 we prove that in the Toda lattice, corresponding to initial data in a ball of radius R/N 2 (0 < R 1) and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Then we consider the original FPU model and prove that, corresponding to the same initial data, energy remains in an exponentially localized packet of Fourier modes for very long times (see Theorem 0.3 below), namely for times one order of magnitude longer then those covered by previous results [BP06]. This is relevant in view of the fact that the time scale covered in [BP06] is that of formation of the packet, so the result that we prove allows to conclude that the packet persists over a time much longer then the one needed for its formation. It is convenient to state the results for Toda and FPU using the small parameter µ := N1 as in [BP06]. We prove the following theorem Theorem 0.2. Consider the Toda lattice (1). Fix σ > 0, then there exist constants R0 , C1 , such that the following holds true. Consider an initial datum with E1 (0) = EN −1 (0) = R2 e−2σ µ4 , Ek (0) ≡ Ek (t) t=0 = 0 , ∀k = 6 1, N − 1 (7) with R < R0 . Then, along the corresponding solution, one has Ek (t) ≤ R2 (1 + C1 R)µ4 e−2σk ,

∀ 1 ≤ k ≤ bN/2c ,

∀t ∈ R .

(8)

For the FPU model we have the following theorem Theorem 0.3. Consider the FPU system (6). Fix s ≥ 1 and σ ≥ 0; then there exist constants R00 , C2 , T, such that the following holds true. Consider a real initial datum fulfilling (7) with R < R00 , then, along the corresponding solution, one has Ek (t) ≤

16R2 µ4 e−2σk , k 2s

∀ 1 ≤ k ≤ bN/2c ,

|t| ≤

Furthermore, for 1 ≤ k ≤ N − 1, consider the action Ik := be its evolution according to the FPU flow. Then one has N −1 1 X 2(s−1) 2σ[k]N [k]N e ω N

k N

T R2 µ4

2 x2k +yk 2

|Ik (t) − Ik (0)| ≤ C3 R2 µ5

·

1 . |β − 1| + C2 Rµ2

(9)

of the Toda lattice and let Ik (t)

for t fullfilling (9)

(10)

k=1

Let us remark that our analysis is part of a project aiming at studying the dynamics of periodic Toda lattices with a large number of particles, in particular its asymptotics. First results in this project were obtained in the papers [BKP09, BKP13b, BKP13a] (see also [BGPU03]). 8

They are based on the Lax pair representation of the Toda lattice in terms of periodic Jacobi matrices. The spectrum of these matrices leads to a complete set of conserved quantities and hence determines the Toda Hamiltonian and the dynamics of Toda lattices, such as their frequencies. In order to study the asymptotics of Toda lattices for a large number N of particles one therefore needs to work in two directions: on the one hand one has to study the asymptotics of the spectrum of Jacobi matrices as N → ∞ and on the other hand, one needs to use tools of the theory of integrable systems in order to effectively extract information on the dynamics of Toda lattices from the periodic spectrum of periodic Jacobi matrices. The KdV on R. In the second part of the thesis we show that for the KdV on the line, the scattering map is an analytic perturbation of the Fourier transform by a 1-smoothing nonlinear operator. With the application we have in mind, we choose a setup for the scattering map so that the spaces considered are left invariant under the KdV flow. Recall that the KdV equation on R ∂t u(t, x) = −∂x3 u(t, x) − 6u(t, x)∂x u(t, x) ,

u(0, x) = q(x) ,

(11)

is globally in time well-posed in various function spaces such as the Sobolev spaces H N ≡ H N (R, R), N ∈ Z≥2 , as well as on the weighted spaces H 2N ∩ L2M , with integers R ∞N ≥ M ≥ 1 [Kat66], where L2M ≡ L2M (R, C) denotes the space of functions satisfying kqk2L2 := −∞ (1 + |x|2 )M |q(x)|2 dx < ∞. M Our analysis relies on a detailed study of the spectral data of the Schrödinger operator L(q) := −∂x2 + q. Denote by f1 (q, x, k) and f2 (q, x, k) the Jost solutions, i.e. solutions of L(q)f = k 2 f with asymptotics f1 (q, x, k) ∼ eikx , x → ∞, f2 (q, x, k) ∼ e−ikx , x → −∞. The eigenvalues of the operator L(q) are called bound states, and a potential q will be said to be without bound states if L(q) has no eigenvalues. Furthermore q will be said to be generic if the Wronskian W (q, k) := [f2 (q, x, k), f1 (q, x, k)] satisfies the condition W (q, 0) 6= 0 (see [Fad64]). We are interested in the analytic properties of the scattering map S(q, k) := [f1 (q, x, k), f2 (q, x, −k)] . which is known to linearize the KdV flow [GGKM74]. To state our result, introduce the set Q := q : R → R , q ∈ L24 : q without bound states and generic ,

(12)

and for any integers N ≥ 0 and M ≥ 4 define QN,M := Q ∩ H N ∩ L2M . We prove that for potentials q ∈ Q, the scattering map S(q, ·) takes value in the space S of functions σ : R → C satisfying (S1) σ(−k) = σ(k),

∀k ∈ R;

(S2) σ(0) > 0. M M Denote by S M,N := S ∩ Hζ,C ∩ L2N . Here Hζ,C is the space of functions f ∈ HCM −1 such that the M th derivative fulfills ζ∂kM f ∈ L2 , where ζ : R → R is an odd monotone C ∞ function with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1 . R +∞ Moreover let F± be the Fourier transformations defined by F± (f ) = −∞ e∓2ikx f (x) dx. In this setup, the scattering map S has the following properties:

Theorem 0.4. For any integers N ≥ 0, M ≥ 4, the following holds: 9

(i) The map S : QN,M → S M,N ,

q 7→ S(q, ·)

is a real analytic diffeomorphism. −1 (ii) The maps A := S − F− and B := S −1 − F− are 1-smoothing, i.e.

A : QN,M → HζM ∩ L2N +1

and

B : S M,N → H N +1 ∩ L2M −1 .

Furthermore they are real analytic maps. Item (i) of Theorem 0.4 shows that the scattering map behaves like a nonlinear Fourier transform, interchanging the decaying and regularity properties. Item (ii) shows that the difference A of the scattering map and its linear part F− is 1-smoothing. Kappeler and Trubowitz [KT86, KT88] studied analytic properties of the scattering map S between weighted Sobolev spaces. More precisely, define the spaces H n,α := f ∈ L2 : xβ ∂xj f ∈ L2 , 0 ≤ j ≤ n, 0 ≤ β ≤ α , H]n,α := f ∈ H n,α : xβ ∂xn+1 f ∈ L2 , 1 ≤ β ≤ α . In [KT86], Kappeler and Trubowitz showed that the map q 7→ S(q, ·) is a real analytic diffeomorphism from Q ∩ H N,N to S ∩ H]N −1,N , N ∈ Z≥3 . They extend their results to potentials with finitely many bound states in [KT88]. Unfortunately, such spaces are not invariant for the KdV flow, thus they are not suited for analyzing qualitative properties of the KdV dynamic. The novelty of our work is to extend the construction of [KT86] to spaces of the form H N ∩ L2M , which, for N ≥ 2M ≥ 2, are invariant for the KdV [Kat66]. As an application of Theorem 0.4 we compare solutions of (11) to solutions of the Cauchy problem for the Airy equation on R, ∂t v(t, x) = −∂x3 v(t, x) ,

v(0, x) = p(x) .

(13)

t t Denote the flows of (13) and (11) by UAiry (p) := v(t, ·) respectively UKdV (q) := u(t, ·). We show N,M t t that for q ∈ Q with N ≥ 2M ≥ 8, the difference UKdV (q) − UAiry (q) is 1-smoothing, i.e. it takes values in H N +1 . More precisely we prove the following:

Theorem 0.5. Let N , M be integers with N ≥ 2M ≥ 8. Then the following holds true: (i) QN,M is invariant under the KdV flow. t t (ii) For any q ∈ QN,M the difference UKdV (q) − UAiry (q) takes values in H N +1 ∩ L2M . Moreover the map

QN,M × R≥0 →H N +1 ∩ L2M ,

t t (q, t) 7→ UKdV (q) − UAiry (q)

is continuous and for any fixed t real analytic in q. This result is motivated from the study of the 1-smoothing property of the KdV flow in the periodic set-up, established recently in [ET13a, KST13] and addresses the question if similar results hold for the KdV flow on the line. In particular in [KST13] the 1-smoothing property of the Birkhoff

10

t t map has been exploited to prove that for q ∈ H N (T, R), N ≥ 1, the difference UKdV (q) − UAiry (q) N +1 is bounded in H (T, R) with a bound which grows linearly in time.

Organization of the thesis. In Chapter 1 we analyze the Toda Lattice with a large number of particles, and we prove Theorem 0.1, Theorem 0.2 and Theorem 0.3. The results of this Chapter are taken from our paper [BM14]. In Chapter 2 we analyze the KdV on R and we prove Theorem 0.5 and Theorem 0.4. The results of this chapter are taken from our paper [MS14]. Each chapter here is self contained and can be read separately.

11

Chapter 1

Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU 1

Introduction and main result

It is well known that the Toda lattice, namely the system with Hamiltonian N −1 N −1 1 X 2 X qj −qj+1 HT oda (p, q) = p + e , 2 j=0 j j=0

(1.1)

and periodic boundary conditions qN = q0 , pN = p0 , is integrable [Tod67, Hén74]. Thus, by standard Arnold-Liouville theory the system admits action angle coordinates. However the actual introduction of such coordinates is quite complicated (see [FM76, FFM82]) and the corresponding transformation has only recently been studied analytically in a series of papers by Henrici and Kappeler [HK08b, HK08c]. In particular such authors have proved the existence of global Birkhoff coordinates, namely canonical coordinates (xk , yk ) analytic on the whole R2N , with the property that the k th action is given by (x2k +yk2 )/2. The construction of Henrici and Kappeler, however is not uniform in the size of the chain, in the sense that the map ΦN introducing Birkhoff coordinates is globally analytic for any fixed N , but it could (and actually does) develop singularities as N → +∞. Here we prove some analyticity properties fulfilled by ΦN uniformly in the limit N → +∞. Precisely we consider complex balls centered at the origin and prove that ΦN maps analytically a ball of radius R/N α in discrete Sobolev-analytic norms into a ball of radius R0 /N α , with R, R0 > 0 independent of N if and only if α ≥ 2. Furthermore we prove that the supremum of ΦN over a complex ball of radius R/N α diverges as N → +∞ when α < 1. In order to prove upper estimates on ΦN we apply to the Toda lattice a Vey type theorem [Vey78] for infinite dimensional systems recently proved by Kuksin and Perelman [KP10]. Actually, we need to prove a new quantitative version of Kuksin-Perelman’s theorem. We think that such a result could be interesting in itself. The lower estimates on the size of ΦN are proved by constructing explicitly the first term of the 12

Taylor expansion of ΦN through Birkhoff normal form techniques; in particular we prove that the second differential d2 ΦN (0) at the origin diverges like N 2 . We finally apply the result to the problem of equipartition of energy in the spirit of FermiPasta-Ulam. We prove that in the Toda lattice, corresponding to initial data with energy E/N 3 (0 < E 1) and with only the first Fourier mode excited, the energy remains forever in a packet of Fourier modes exponentially decreasing with the wave number. Then we consider the original FPU model and prove that, corresponding to the same initial data, energy remains in an exponentially localized packet of Fourier modes for times of order N 4 (see Theorem 1.16 below), namely for times one order of magnitude longer then those covered by previous results (see [BP06], see also [SW00, HL12]). This is relevant in view of the fact that the time scale of formation of the packet is N 3 (see [BP06]), so our result allows to conclude that the packet persists over a time much longer then the one needed for its formation.

1.1

Birkhoff coordinates for the Toda lattice

We come to a precise statement of the main results of the present chapter. Consider the Toda lattice in the subspace characterized by X X qj = 0 = pj (1.2) j

j

which is invariant under the dynamics. Introduce the discrete Fourier transform F(q) = qˆ defined by N −1 1 X qj e2iπjk/N , k∈Z, (1.3) qˆk = √ N j=0 and consider pˆk defined analogously. Due to (1.2) one has pˆ0 = qˆ0 = 0 and furthermore pˆk = N −1 pˆk+N , qˆk = qˆk+N , ∀k ∈ Z, so we restrict to {ˆ pk , qˆk }k=1 . Corresponding to real sequences (pj , qj ) one has qˆk = qˆN −k and pˆk = pˆN −k . Introduce the linear Birkhoff variables pˆk + pˆN −k − iωk (ˆ qk − qˆN −k ) pˆk − pˆN −k + iωk (ˆ qk + qˆN −k ) √ √ , Yk = , k = 1, ..., N − 1 , 2ωk i 2ωk (1.4) k where ωk ≡ ω N := 2 sin(kπ/N ); using such coordinates, which are symplectic, the quadratic part N −1 2 X pj + (qj − qj+1 )2 H0 := (1.5) 2 j=0 Xk =

of the Hamiltonian takes the form H0 =

N −1 X

ω

k N

k=1

Xk2 + Yk2 . 2

(1.6)

With an abuse of notations, we re-denote by HT oda the Hamiltonian (1.1) written in the coordinates (X, Y ). The following theorem is due to Henrici and Kappeler: 13

Theorem 1.1 ([HK08c]). For any integer N ≥ 2 there exists a global real analytic symplectic diffeomorphism ΦN : RN −1 × RN −1 → RN −1 × RN −1 , (X, Y ) = ΦN (x, y) with the following properties: (i) The Hamiltonian HT oda ◦ ΦN is a function of the actions Ik := Birkhoff variables for the Toda Lattice.

2 x2k +yk 2

only, i.e. (xk , yk ) are

(ii) The differential of ΦN at the origin is the identity: dΦN (0, 0) = 1. Our main results concern the analyticity properties of the map ΦN as N → ∞. To come to a precise statement we have to introduce a suitable topology in CN −1 × CN −1 . For any s ≥ 0, σ ≥ 0 introduce in CN −1 × CN −1 the discrete Sobolev-analytic norm 2

k(X, Y )kP s,σ :=

N −1 1 X 2s 2σ[k]N ω [k]N e N k=1

k N

|Xk |2 + |Yk |2 2

(1.7)

where [k]N := min(|k|, |N − k|) . N −1

N −1

The space C ×C endowed by such a norm will be denoted by P s,σ . We denote by B s,σ (R) the ball of radius R and center 0 in the topology defined by the norm k.kP s,σ . We will also denote by BRs,σ := B s,σ (R) ∩ (RN −1 × RN −1 ) the real ball of radius R. Remark 1.2. When σ = s = 0 the norm (1.7) coincides with the energy norm rescaled by a factor 1/N (the rescaling factor will be discussed in Remark 1.11). We are particularly interested in the case σ > 0 since, in such a case, states belonging to P s,σ are exponentially decreasing in Fourier space. The consideration of positive values of s will be needed in the proof of the main theorem. Our main result is the following Theorem. Theorem 1.3. For any s ≥ 0, σ ≥ 0 there exist strictly positive constants Rs,σ , Cs,σ , such that for any N ≥ 2, the map ΦN is analytic as a map from B s,σ (Rs,σ /N 2 ) to P s,σ and fulfills sup k(x,y)kP s,σ ≤R/N 2

kΦN (x, y) − (x, y)kP s+1,σ ≤ Cs,σ

R2 , N2

∀R < Rs,σ .

(1.8)

The same estimated is fulfilled by the inverse map Φ−1 N possibly with a different Rs,σ . Remark 1.4. The estimate (1.8) controls the size of the nonlinear corrections in a norm which is stronger then the norm of (x, y), showing that ΦN − 1 is 1-smoothing. The proof of this kind of smoothing effect was actually the main aim of the work by Kuksin and Perelman [KP10], which proved it for KdV. Subsequently Kappeler, Schaad and Topalov [KST13] proved that such a smoothing property holds also globally for the KdV Birkhoff map. Remark 1.5. As a consequence of (1.8) one has R R s,σ s,σ ⊂B (1 + Cs,σ R) , ΦN B N2 N2

∀R < Rs,σ , ∀N ≥ 2

and the same estimate is fulfilled by the inverse map Φ−1 N , possibly with a different Rs,σ . 14

(1.9)

Corollary 1.6. For any s ≥ 0, σ ≥ 0 there exist strictly positive constants Rs,σ , Cs,σ , with the following property. Consider the solution v(t) ≡ (X(t), Y (t)) of the Toda Lattice corresponding to initial data v0 ∈ B s,σ NR2 with R ≤ Rs,σ then one has R s,σ (1 + Cs,σ R) , ∀t ∈ R . (1.10) v(t) ∈ B N2 In order to state a converse of Theorem 1.3 consider the second differential QΦN := d2 ΦN (0, 0) of ΦN at the origin; QΦN : P s,σ → P s,σ is a quadratic polynomial in the phase space variables1 . Theorem 1.7. For any s ≥ 0, σ ≥ 0 there exist strictly positive R, C, Ns,σ ∈ N, such that, for any N ≥ Ns,σ , α ∈ R, the quadratic form QΦN fulfills

Φ

Q N (v, v) s,σ ≥ CR2 N 2−2α . (1.11) sup P v∈BRs,σ ( NRα ) Remark 1.8. Roughtly speaking, one can say that, as N → ∞, the real diffeomorphism ΦN develops a singularity at zero in the second derivative. Using Cauchy estimate (see subsect. 3.2) one immediately gets the following corollary. Corollary 1.9. Assume that for some s ≥ 0, σ ≥ 0 there exist strictly positive R, R0 and α ≥ 0, α0 ∈ R, Ns,σ ∈ N, s.t., for any N ≥ Ns,σ , the map ΦN is analytic in the complex ball B s,σ (R/N α ) and fulfills 0 R R s,σ s,σ ΦN B , (1.12) ⊂B α N N α0 then one has α0 ≤ 2(α − 1). Remark 1.10. A particular case of Corollary 1.9 is α < 1, in which one has that the image of a ball of radius RN −α under ΦN is unbounded as N → ∞. A further interesting case is that of α = α0 , which implies α ≥ 2, thus showing that the scaling R/N 2 is the best possible one in which a property of the kind of (1.9) holds. Remark 1.11. A state (X, Y ) is in the ball B s,σ (R/N 2 ) if and only if there exist interpolating periodic functions (β, α), namely functions s.t. j j pj = β , qj − qj+1 = α , (1.13) N N which are analytic in a strip of width σ and have a Sobolev-analytic norm of size R/N 2 . More precisely, given a state (p, q) one considers its Fourier coefficients (ˆ p, qˆ) and the corresponding X, Y variables; define N −1 1 X qˆk 1 − e−2πik/N e−2πixk , α(x) = √ N k=0 1 actually

N −1 1 X β(x) = √ pˆk e−2πixk N k=0

according to the estimate (1.8) it is smooth as a map P s,σ → P s+1,σ

15

which fulfill (1.13). Then the Sobolev-analytic norms of α and β are controlled by k(X, Y )kP s,σ . For example one has 2

2

2

k(α, β)kH s := kαkL2 + kβkL2 +

1 1 2 2 2 k∂ s αk 2 + k∂ s βk 2 = k(X, Y )kP s,0 , (2π)2s x L (2π)2s x L

R1 2 2 where kαkL2 := 0 |α(x)| dx. In particular we consider here states with Sobolev-analytic norm 2 of order R/N with R 1. The factor 1/N in the definition of the norm was introduced to get correspondence between the norm of a state and the norm of the interpolating functions. Remark 1.12. As a consequence of Remark 1.11, the order in N of the solutions we are describing with Theorem 1.3 is the same of the solutions studied in the papers [BP06] and [BKP09, BKP13b, BKP13a]. Remark 1.13. The results of Theorem 1.3 and Theorem 1.7 extend to states with discrete SobolevGevrey norm defined by k(X, Y

2 )kP s,σ,ν

N −1 1 X 2s 2σ[k]νN [k]N e ω := N k=1

k N

|Xk |2 + |Yk |2 2

(1.14)

where 0 ≤ ν ≤ 1. As a consequence of Remark 1.11, these states are interpolated by periodic functions with regularity Gevrey ν. Our analysis is part of a project aiming at studying the dynamics of periodic Toda lattices with a large number of particles, in particular its asymptotics. First results in this project were obtained in the papers [BKP09, BKP13b, BKP13a]. They are based on the Lax pair representation of the Toda lattice in terms of periodic Jacobi matrices. The spectrum of these matrices leads to a complete set of conserved quantities and hence determines the Toda Hamiltonian and the dynamics of Toda lattices, such as their frequencies. In order to study the asymptotics of Toda lattices for a large number N of particles one therefore needs to work in two directions: on the one hand one has to study the asymptotics of the spectrum of Jacobi matrices as N → ∞ and on the other hand, one needs to use tools of the theory of integrable systems in order to effectively extract information on the dynamics of Toda lattices from the periodic spectrum of periodic Jacobi matrices. The limit of a class of sequences of N × N Jacobi matrices as N → ∞ has been formally studied already at the beginning of the theory of the Toda lattices (see e.g. [Tod67]). However, as pointed out in [BKP13b], these studies only allowed to (formally) compute the asymptotics of the spectrum in special cases. In particular, Toda lattices, which incorporated right and left moving waves could not be analyzed at all in this way. In [BKP13b], based on an approach pioneered in [BGPU03], the asymptotics of the spectra of sequences of Jacobi matrices corresponding to states of the form (1.13) were rigorously derived by the means of semiclassical analysis. It turns out that in such a limit the spectrum splits into three parts: one group of eigenvalues at each of the two edges of the spectrum within an interval of size O(N −2 ), whose asymptotics are described by certain Hill operators, and a third group of eigenvalues, consisting of the bulk of the spectrum, whose asymptotics coincides with the one of Toda lattices at the equilibrium – see [BKP13b] for details. In [BKP13a] the asymptotics of the eigenvalues obtained in [BKP13b] were used in order to compute the one of the actions and of the frequencies of Toda lattices. In particular it was shown that the asymptotics of the frequencies at the two edges involve the frequencies of two KdV solutions.

16

The tools used in [BKP13a] are those of the theory of infinite dimensional integrable systems as developed in [KP03] and adapted to the Toda lattice in [HK08b]. The present thesis takes up another important topic in the large number of particle limit of periodic Toda lattices: we study the Birkhoff coordinates near the equilibrium in the limit of large N to provide precise estimates on the size of complex balls around the equilibrium in Fourier coordinates and the corresponding size in Birkhoff coordinates. Our analysis allows to describe the evolution of Toda lattices with large number of particles in the original coordinates and to obtain an application to the study of FPU lattices (on which we will comment in the next section). We remark that the obtained estimates on the size of the complex balls are optimal. In our view this is a strong indication that beyond such a regime the standard tools of integrable systems become inadequate for studying the asymptotic features of the dynamics of the periodic Toda lattices as N → ∞. The proofs of our results are based on a novel technique developed in [KP10] to show a Vey type theorem for the KdV equation on the circle which we adapt here to the study of Toda lattices, developing in this way another tool for the study of periodic Toda lattices with a large number of particles. We remark that for our arguments to go through, we need to assume an additional smallness condition on the set of states admitted as initial data: the states are required to be interpolated by functions α and β with Sobolev-analytic norm of size R/N 2 , with R 1 sufficiently small. (In the papers [BKP09, BKP13b, BKP13a], the size R can be arbitrarily large.)

1.2

On the FPU metastable packet

In this subsection we recall the phenomenon of the formation of a packet of modes in the FPU chain and state our related results. First of all we recall that the FPU (α, β)-model is the Hamiltonian lattice with Hamiltonian function which, in suitable rescaled variables, takes the form HF P U (p, q) =

N −1 X j=0

U (x) =

p2j + U (qj − qj+1 ) , 2

x2 x3 x4 + +β . 2 6 24

(1.15) (1.16)

We will consider the case of periodic boundary conditions: q0 = qN , p0 = pN . Remark 1.14. One has HF P U (p, q) = HT oda (p, q) + (β − 1)H2 (q) + H (3) (q), where

H (3)

N −1 X

(qj − qj+1 )l+2 , (l + 2)! j=0 X := − Hl .

Hl (q) :=

∀l ≥ 2 ,

l≥3

Introduce the energies of the normal modes by k 2 |ˆ p k |2 + ω N |ˆ qk |2 Ek := , 2 17

1≤k ≤N −1 ,

(1.17)

correspondingly denote by Ek :=

Ek N

(1.18)

the specific energy in the k th mode. Note that since p, q are real variables, one has Ek = EN −k . In their celebrated numerical experiment Fermi Pasta and Ulam [FPU65], being interested in the problem of foundation of statistical mechanics, studied both the behaviour of Ek (t) and of its time average Z 1 t Ek (s)ds . hEk i(t) := t 0 They observed that, corresponding to initial data with E1 (0) 6= 0 and Ek (0) = 0 ∀k 6= 1, N − 1, the quantities Ek (t) present a recurrent behaviour, while their averages hEk i(t) quickly relax to a sequence E¯k exponentially decreasing with k. This is what is known under the name of FPU packet of modes. Subsequent numerical observations have investigated the persistence of the phenomenon for large N and have also shown that after some quite long time scale (whose precise length is not yet understood) the averages hEk i(t) relax to equipartition (see e.g. [BGG04, BGP04, BP11, BCP13]). This is the phenomenon known as metastability of the FPU packet. The idea of exploiting the vicinity of FPU with Toda in order to study the dynamics of FPU goes back to [FFM82], in which the authors performed some numerical investigations studying the evolution of the Toda invariants in the dynamics of FPU. A systematic numerical study of the evolution of the Toda invariants in FPU, paying particular attention to the dependence on N of the phenomena, was performed by Benettin and Ponno [BP11] (see also [BCP13]). In particular such authors put into evidence the fact that the FPU packet seems to have an infinite lifespan in the Toda lattice. Furthermore they showed that the relevant parameter controlling the lifespan of the packet in the FPU model is the distance of FPU from the corresponding Toda lattice. Our Theorem 1.3 yields as a corollary the effective existence and infinite persistence of the packet in the Toda lattice and also an estimate of its lifespan in the FPU system, estimate in which the effective parameter is the distance between Toda and FPU. It is convenient to state the results for Toda and FPU using the small parameter µ :=

1 N

as in [BP06]. The following corollary is an immediate consequence of Corollary 1.6. Corollary 1.15. Consider the Toda lattice (1.1). Fix σ > 0, then there exist constants R0 , C1 , such that the following holds true. Consider an initial datum with E1 (0) = EN −1 (0) = R2 e−2σ µ4 , Ek (0) ≡ Ek (t) t=0 = 0 , ∀k 6= 1, N − 1 (1.19) with R < R0 . Then, along the corresponding solution, one has Ek (t) ≤ R2 (1 + C1 R)µ4 e−2σk ,

∀ 1 ≤ k ≤ bN/2c ,

For the FPU model we have the following corollary

18

∀t ∈ R .

(1.20)

Theorem 1.16. Consider the FPU system (1.15). Fix s ≥ 1 and σ ≥ 0; then there exist constants R00 , C2 , T, such that the following holds true. Consider a real initial datum fulfilling (1.19) with R < R00 , then, along the corresponding solution, one has Ek (t) ≤

16R2 µ4 e−2σk , k 2s

∀ 1 ≤ k ≤ bN/2c ,

|t| ≤

T R2 µ4

Furthermore, for 1 ≤ k ≤ N − 1, consider the action Ik := be its evolution according to the FPU flow. Then one has N −1 1 X 2(s−1) 2σ[k]N ω [k]N e N

k N

·

2 x2k +yk 2

|Ik (t) − Ik (0)| ≤ C3 R2 µ5

1 . |β − 1| + C2 Rµ2

(1.21)

of the Toda lattice and let Ik (t)

for t fullfilling (1.21)

(1.22)

k=1

Remark 1.17. The estimates (1.21) are stronger then the corresponding estimates given in [BP06], which are T Ek (t) ≤ C1 µ4 e−σk + C2 µ5 , ∀ 1 ≤ k ≤ bN/2c , |t| ≤ 3 . µ First, the time scale of validity of (1.21) is one order longer than that of [BP06]. Second we show that as β approaches the value corresponding to the Toda lattice (1 in our units) the time of stability improves. Third the exponential estimate of Ek as a function of k is shown to hold also for large values of k (the µ5 correction is missing). Finally in [BP06] it was shown that T /µ3 is the time of formation of the metastable packet. So we can now conclude that the time of persistence of the packet is at least one order of magnitude larger (namely µ−4 ) with respect to the time needed for its formation. Remark 1.18. We recall also the result of [HL12] in which the authors obtained a control of the dynamics for longer time scales, but for initial data with much smaller energies. Remark 1.19. Recently some results on energy sharing in FPU in the thermodynamic limit [MBC14](see also [Car07, CM12, GPP12]) have also been obtained, however such results are not able to explain the formation and the stability of the FPU packet of modes.

2 2.1

A quantitative Kuksin-Perelman Theorem Statement of the theorem

In this section we state and prove a quantitative version of Kuksin-Perelman Theorem which will be used to prove Theorem 1.3. It is convenient to formulate it in the framework of weighted `2 spaces, that we are going now to recall. 2 For any N ≤ ∞, given a sequence w = {wk }N k=1 , wk ≥ 1 ∀k ≥ 1, consider the space `w of complex sequences ξ = {ξk }N with norm k=1 2 kξkw

:=

N X

wk2 |ξk |2 < ∞.

(1.23)

k=1 2

Denote by P w the complex Banach space P w := `2w ⊕`2w 3 (ξ, η) endowed with the norm k(ξ, η)kw := 2 2 kξkw + kηkw . We denote by PRw the real subspace of P w defined by PRw := (ξ, η) ∈ P w : ηk = ξ k ∀ 1 ≤ k ≤ N . (1.24) 19

We will denote by B w (ρ) (respectively BRw (ρ)) the ball in the topology of P w (respectively PRw ) with center 0 and radius ρ > 0. Remark 1.20. In the case of the Toda lattice the variables (ξ, η) are defined by k k pˆk + iω N pˆN −k − iω N qˆk qˆN −k q ξk = q , ηk = , 1≤k ≤N −1 , k k 2ω N 2ω N

(1.25)

and their connection with the real Birkhoff variables is given by Xk =

ξk + η k √ , 2

Yk =

ξk − ηk √ , i 2

1≤k ≤N −1 .

(1.26)

We denote by P 1 the Banach space of sequences in which all the weights wk are equal to 1. For X , Y Banach spaces, we shall write L(X , Y) to denote the set of linear and bounded operators from X to Y. For X = Y we will write just L(X ). Remark 1.21. In the application to the Toda lattice with N particles we will use a finite, but not k 2 3 2s 2σk fixed N and weights of the form wk2 = wN ω N , 1 ≤ k ≤ bN/2c. −k = N k e Given two weights w1 and w2 , we will say that w1 ≤ w2 iff wk1 ≤ wk2 , ∀k. Sometimes, when there is no risk of confusion, we will omit the index w from the different quantities. In P 1 we will use the scalar product

N X 2 (ξ 1 , η 1 ), (ξ 2 , η 2 ) c := ξk1 ξ k + ηk1 η 2k .

(1.27)

k=1

Correspondingly, the scalar product and symplectic form on the real subspace PRw are given for ξ 1 ≡ (ξ 1 , ξ¯1 ) and ξ 2 ≡ (ξ 2 , ξ¯2 ) by N X

1 2 ξ , ξ := 2Re ξk1 ξk2 ,

ω0 (ξ 1 , ξ 2 ) := E ξ 1 , ξ 2 ,

(1.28)

k=1

where E := −i. Given a smooth F : PRw → C, we denote by XF the Hamiltonian vector field of F , given by XF = J∇F , where J = E −1 . For F, G : PRw → C we denote by {F, G} the Poisson bracket (with respect to ω0 ): {F, G} := h∇F, J∇Gi (provided it exists). We say that the functions F, G commute if {F, G} = 0. In order to state the main abstract theorem we start by recalling the notion of normally analytic map, exploited also in [Nik86] and [BG06]. First we recall that a map P˜ r : (P w )r → B, with B a Banach space, is said to be r-multilinear if P˜ r (v (1) , . . . , v (r) ) is linear in each variable v (j) ≡ (ξ (j) , η (j) ); a r-multilinear map is said to be bounded if there exists a constant C > 0 such that

˜ r (1)

∀v (1) , . . . , v (r) ∈ P w .

P (v , . . . , v (r) ) ≤ C v (1) . . . v (r) B

w

20

w

Correspondingly its norm is defined by

˜r

˜ r (1)

sup

P :=

P (v , · · · , v (r) ) . B (1) (r) kv kw ,··· ,kv kw ≤1 A map P r : P w → B is a homogeneous polynomial of order r if there exists a r-multilinear map P˜ r : (P w )r → B such that P r (v) = P˜ r (v, . . . , v) ∀v ∈ P w . (1.29) A r- homogeneous polynomial is bounded if it has finite norm kP r k := sup kP r (v)kB . kvkw ≤1

Remark 1.22. Clearly kP r k ≤ P˜ r . Furthermore one has P˜ r ≤ er kP r k – cf. [Muj86]. It is easy to see that a multilinear map and the corresponding polynomial are continuous (and analytic) if and only if they are bounded. Let P r : P w → B be a homogeneous polynomial of order r; assume B separable and let {bn }n≥1 ⊂ B be a basis for the space B. Expand P r as follows X r,n K L P r (v) ≡ P r (ξ, η) = PK,L ξ η bn , (1.30) |K|+|L|=r n≥1 KN K where K, L ∈ NN ≡ ξ1K1 · · · ξN , 0 , N0 = N ∪ {0}, |K| := K1 + · · · + KN , ξ ≡ {ξj }j≥1 and ξ LN η L ≡ η1L1 · · · ηN .

Definition 1.23. The modulus of a polynomial P r is the polynomial P r defined by X r,n P r (ξ, η) := PK,L ξ K η L bn .

(1.31)

|K|+|L|=r n≥1

A polynomial P r is said to have bounded modulus if P r is a bounded polynomial. A map F : P w → B is said to be an analytic germ if there exists ρ > 0 such that F : B w (ρ) → B is analytic. F can be written as a power series absolutely and uniformly convergent in B w (ρ): P Then r F (v) = r≥0 F (v). Here F r (v) is a homogeneous polynomial of degree r in the variables v = (ξ, η). We will write F = O(v n ) if in the previous expansion F r (v) = 0 for every r < n. Definition 1.24. An analytic germ F : P w → B is said to be normally analytic if there exists ρ > 0 such that X F (v) := F r (v) (1.32) r≥0 w

is absolutely and uniformly convergent in B (ρ). In such a case we will write F ∈ Nρ (P w , B). Nρ (P w , B) is a Banach space when endowed by the norm |F |ρ :=

sup

kF (v)kB .

(1.33)

v∈B w (ρ)

Let U ⊂ PRw be open. A map F : U → B is said to be a real analytic germ (respectively real normally analytic) on U if for each point u ∈ U there exist a neighborhood V of u in P w and an analytic germ (respectively normally analytic germ) which coincides with F on U ∩ V . 21

Remark 1.25. It follows from Cauchy inequality that the Taylor polynomials F r of F satisfy r

kF r (v)kB ≤ |F |ρ

kvkw ρr

∀v ∈ B w (ρ) .

(1.34)

Remark 1.26. Since ∀r ≥ 1 one has kF r k ≤ kF r k , if F ∈ Nρ (P w , B) then the Taylor series of F is uniformly convergent in B w (ρ). The case B = P w will be of particular importance; in this case the basis {bj }j≥1 will coincide with the natural basis {ej }j≥1 of such a space (namely the vectors with all components equal to 1 2 zero except the j th one which is equal to 1). We will consider also the case B = L(P w , P w ) 1 2 (bounded linear operators from P w to P w ), where w1 and w2 are weights. Here the chosen basis is bjk = ej ⊗ ek (labeled by 2 indexes). Remark 1.27. For v ≡ (ξ, η) ∈ P 1 , we denote by |v| the vector of the modulus of the com1 2 ponents of v: |v| = (|v1 |, . . . , |vN |), |vj | := (|ξj |, |ηj |). If F ∈ Nρ (P w , P w ) then dF (|v|)|u| ≤ dF (|v|)|u| (see [KP10]) and therefore, for any 0 < d < 1, Cauchy estimates imply that dF ∈ 1 1 2 N(1−d)ρ (P w , L(P w , P w )) with 1 |F |ρ , (1.35) |dF |ρ(1−d) ≤ dρ where dF is computed with respect to the basis ej ⊗ ek . Following Kuksin-Perelman [KP10] we will need also a further property. 1

2

2

Definition 1.28. A normally analytic germ F ∈ Nρ (P w , P w ) will be said to be of class Aw w1 ,ρ if 1

1

2

F = O(v 2 ) and the map v 7→ dF (v)∗ ∈ Nρ (P w , L(P w , P w )). Here dF (v)∗ is the adjoint operator 2 of dF (v) with respect to the standard scalar product (1.27). On Aw w1 ,ρ we will use the norm kF kAw2

w1 ,ρ

:= |F |ρ + ρ |dF |ρ + ρ |dF ∗ |ρ .

(1.36)

2

Remark 1.29. Assume that for some ρ > 0 the map F ∈ Aw w1 ,ρ , then for every 0 < d ≤ 2 2 |F |dρ ≤ 2d |F |ρ and kF kAw2 ≤ 6d kF kAw2 . w1 ,dρ

1 2

one has

w1 ,ρ

2

1

1

2

A real normally analytic germ F : BRw (ρ) → PRw will be said to be of class Nρ (PRw , PRw ) 2 2 2 w1 (respectively Aw , P w ) (respectively Aw w1 ,ρ ) if there exists a map of class Nρ (P w1 ,ρ ), which coincides 1

with F on BRw (ρ). In this case we will also denote by |F |ρ (respectively kF kAw2 ) the norm defined w1 ,ρ

by (1.33) (respectively (1.36)) of the complex extension of F . 1 2 Let now F : U ⊂ P w → P w be an analytic map. We will say that F is real for real sequences 1 2 ¯ = F2 (ξ, ξ). ¯ Clearly, the if F (U ∩ PRw ) ⊆ PRw , namely F (ξ, η) = (F1 (ξ, η), F2 (ξ, η)) satisfies F1 (ξ, ξ) restriction F |U ∩P w1 is a real analytic map. R

We come now to the statement of the Vey Theorem. 1 1 2 Fix ρ > 0 and let Ψ : BRw (ρ) → PRw , Ψ = 1 + Ψ0 with 1 the identity map and Ψ0 ∈ Aw w1 ,ρ . Write Ψ component-wise, Ψ = (Ψj , Ψj ) j≥1 , and consider the foliation defined by the functions

22

n o 2 |Ψj (v)| /2

j≥1

. Given v ∈ PRw we define the leaf through v by Fv :=

u ∈ PRw :

|Ψj (v)|2 |Ψj (u)|2 = , ∀j ≥ 1 2 2

.

(1.37)

S Let F = v∈P w Fv be the collection of all the leaves of the foliation. We will denote by Tv F R the tangent space to Fv at the point v ∈ PRw . A relevant role will also be played by the function I = {Ij }j≥1 whose components are defined by 2 ¯ := |ξj | Ij (v) ≡ Ij (ξ, ξ) 2

∀j ≥ 1 .

(1.38)

The foliation they define will be denoted by F (0) . Remark 1.30. Ψ maps the foliation F into the foliation F (0) , namely F (0) = Ψ(F). The main theorem of this section is the following Theorem 1.31. (Quantitative version of Kuksin-Perelman Theorem) Let w1 and w2 be weights 1 with w1 ≤ w2 . Consider the space PRw endowed with the symplectic form ω0 defined in (1.28). Let 1 1 2 ρ > 0 and assume Ψ : BRw (ρ) → PRw , Ψ = 1 + Ψ0 and Ψ0 ∈ Aw w1 ,ρ . Define

1 := Ψ0 Aw2 . (1.39) w1 ,ρ

2

Assume that the functionals { 21 |Ψj (v)| }j≥1 pairwise commute with respect to the symplectic form ω0 , and that ρ is so small that 1 < 2−34 ρ. (1.40) e : B w1 (aρ) → P w1 , a = 2−48 , with the following Then there exists a real normally analytic map Ψ R R properties: e ∗ ω0 = ω0 , so that the coordinates z := Ψ(v) e i) Ψ are canonical; 2 e ii) the functionals 12 Ψ pairwise commute with respect to the symplectic form ω0 ; j (v) j≥1

e e iii) F (0) = Ψ(F), namely the foliation defined by Ψ coincides with the foliation defined by Ψ;

e 0 e =1+Ψ e 0 with Ψ e 0 ∈ Aw21 iv) Ψ ≤ 217 1 . w ,aρ and Ψ w2 A

w1 ,aρ

The following corollary holds: 1

Corollary 1.32. Let H : PRw → R be a real analytic Hamiltonian function. Let Ψ be as in Theorem 2 1.31 and assume that for every j ≥ 1, |Ψj (v)| is an integral of motion for H, i.e. {H, |Ψj |2 } = 0

∀ j ≥ 1.

(1.41)

e j (v) are real Birkhoff coordinates for H, namely Then the coordinates (xj , yj ) defined by xj +iyj = Ψ canonical conjugated coordinates in which the Hamiltonian depends only on (x2j + yj2 )/2. 23

Proof of Corollary 1.32. Since Ψ = 1 + Ψ0 , the functions Ψj (v) can be used as coordinates in a e be the map in the statement of Theorem 1.31. Denote suitable neighborhood of 0 in PRw . Let Ψ 2 1 e Fl (v) := Ψl (v) . Since the foliation defined by the functions {Fl }l≥1 and the foliation defined 2

2

2

by {|Ψj | }j≥1 coincide (Theorem 1.31 iii)), each Fl is constant on the level sets of {|Ψj | }j≥1 . It 2 2 follows that each Fl is a function of {|Ψj | }j≥1 only. Since ∀ j ≥ 1, |Ψj | is an integral of motion for H, the same is true for Fl , ∀l ≥ 1. Define now, in a suitable neighborhood of the origin, the 2 e j , z¯j ≡ Ψ e j . Of course Fl = |zl | . By (1.41) it follows then that coordinates (z, z¯) by zj ≡ Ψ 2 ∂H 1 ∂H zl − z¯l . (1.42) 0 = {H, zl z¯l } = i ∂zl ∂ z¯l e e is invertible and its inverse Φ e satisfies Φ e = 1+Φ e 0 with Since dΨ(0) = 1 (Theorem 1.31 iv)),

Ψ

2

18 0 0 w 0 e e ∈A 1 e ≤ 2 1 (Lemma 1.65 ii) in Appendix A). ≤ 2 Ψ Φ w ,aµρ and Φ w2 w2 A

w1 ,aµρ

A

w1 ,aρ

e in Taylor series in the variables (z, z¯): Expand now H ◦ Φ X r e z¯) = H ◦ Φ(z, Hα,β z α z¯β . r≥2, |α|+|β|=r

e is a Then equation (1.42) implies that in each term of the summation α = β, therefore H ◦ Φ function of |z1 |2 , . . . , |zN |2 . Define now the real variables (x, y) as in the statement, then the claim follows immediately.

2.2

Proof of the Quantitative Kuksin-Perelman Theorem

In this section we recall and adapt Eliasson’s proof [Eli90] of the Vey Theorem following [KP10]. As we anticipated in the introduction, the novelty of our approach is to add quantitative estimates on e of Theorem 1.31. In Appendix A we show that the class of normally analytic the Birkhoff map Ψ maps is closed under several operations like composition, inversion and flow-generation, and provide new quantitative estimates which will be used during the proof below. The idea of the proof of Theorem 1.31 is to consider the functions {Ψj (v)}j≥1 as noncanonical coordinates, and to look for a coordinate transformation introducing canonical variables and preserving the foliation F (0) (which is the image of F in the noncanonical variables). This will be done in two steps both based on the standard procedure of Darboux Theorem that we now recall. In order to construct a coordinate transformation ϕ transforming the closed nondegenerate form Ω1 into a closed nondegenerate form Ω0 , then it is convenient to look for ϕ as the time 1 flow ϕt of a time-dependent vector field Y t . To construct Y t one defines Ωt := Ω0 +t(Ω1 −Ω0 ) and imposes that d 0 = dt ϕt∗ Ωt = ϕt∗ (LY t Ωt + Ω1 − Ω0 ) = ϕt∗ d(Y t yΩt ) + d(α1 − α0 ) t=0 where α1 , α0 are potential forms for Ω1 and Ω0 (namely dαi = Ωi , i = 0, 1) and LY t is the Lie derivative of Y t . Then one gets Y t yΩt + α1 − α0 = df (1.43) 24

for each f smooth; then, if Ωt is nondegenerate, this defines Y t . If Y t generates a flow ϕt defined up to time 1, the map ϕ := ϕt |t=1 satisfies ϕ∗ Ω1 = Ω0 . Thus, given Ω0 and Ω1 , the whole game reduces to study the analytic properties of Y t and to prove that it generates a flow. A non-constant symplectic form Ω will always be represented through a linear skew-symmetric invertible operator E as follows: Ω(v)(u(1) ; u(2) ) = hE(v)u(1) ; u(2) i ,

∀u(1) , u(2) ∈ Tv PRw ' PRw .

(1.44)

We denote by {F, G}Ω the Poisson bracket with respect to Ω : {F, G}Ω := h∇F, J∇Gi, J := E −1 . Similarly we will represent 1-forms through the vector field A such that α(v)(u) = hA(v), ui,

∀u ∈ Tv PRw .

(1.45)

Define ω1 := (Ψ−1 )∗ ω0 , and let Eω1 be the operator representing the symplectic form ω1 . The first step consists in transforming ω1 to a symplectic form whose "average over F (0) " coincides with ω0 . So we start by defining precisely what “average of k-forms” means. To this end consider the 2 Hamiltonian vector fields XI0l of the functions Il ≡ |v2l | through the symplectic form ω0 ; they are given by XI0l (v) = i∇Il (v) = ivl el , ∀ l ≥ 1. (1.46) For every l ≥ 1 the corresponding flow φtl ≡ φtX 0 is given by Il

φtl (v) = v1 , · · · , vl−1 , eit vl , vl+1 , · · ·

.

Remark that the map φtl is linear in v, 2π periodic in t and its adjoint satisfies (φtl )∗ = φ−t l . Given a k-form α on PRw (k ≥ 0), we define its average by Mj α(v) =

1 2π

Z

2π

Z

((φtj )∗ α)(v)dt,

j≥1,

and

M α(v) =

[(φθ )∗ α] dθ

(1.47)

T

0

where T is the (possibly infinite dimensional) torus, the map φθ = (φθ11 ◦ φθ22 · · · ) and dθ is the Haar measure on T . Remark 1.33. In the particular cases of 1 and 2-forms it is useful to compute the average in term of the representations (1.44) and (1.45). Thus, for v, u(1) , u(2) ∈ PRw , if α(v)u(1) = hA(v); u(1) i ,

ω(v)(u(1) , u(2) ) = hE(v) u(1) ; u(2) i ,

one has (1)

(M α)(v)u

(1)

= h(M A)(v); u

Z i,

with

M A(v) =

φ−θ A(φθ (v)) dθ

(1.48)

T

and (1)

(M ω)(v)(u

(2)

,u

(1)

) = h(M E)(v)u

(2)

;u

Z i , with M E(v) = T

25

φ−θ E(φθ (v))φθ dθ.

(1.49)

Remark 1.34. The operator M commutes with the differential operator d and the rotations φθ . In particular M A(v) and M E(v) as in (1.48), (1.49) satisfy φθ M A(v) = M A(φθ v),

φθ M E(v)u = M E(φθ v)φθ u,

∀θ ∈ T .

We study now the analytic properties of ω1 and of its potential form αω1 . In the rest of the P∞ section denote by S := n=1 1/n2 and by µ := 1/e(32S)1/2 ≈ 0.0507 .

(1.50)

Lemma 1.35. Let Φ := Ψ−1 and ω1 be as above. Assume that 1 ≤ ρ/e. Then the following holds: 1

1

2

(i) Eω1 = −i + Υω1 , with Υω1 ∈ Nµρ (PRw , L(PRw , PRw )) and |Υω1 |µρ ≤

81 . µρ

(1.51)

(ii) Define Z

1

Wω1 (v) :=

Υω1 (tv)tv dt ,

(1.52)

0 2

then Wω1 ∈ Aw w1 ,µ3 ρ and kWω1 kAw2

w1 ,µ3 ρ

≤ 81 . Moreover the 1-form αWω1 := hWω1 ; .i satisfies

dαWω1 = ω1 − ω0 .

0 −1 2

≤ Proof. By Lemma 1.65 one has that Φ = 1 + Ψ0 = 1 + Φ0 with Φ0 ∈ Aw w1 ,µρ and Φ Aw21 w ,µρ

0 2 Ψ Aw2 ≤ 21 . To prove (i), just remark that w1 ,ρ

Eω1 (v) = dΦ∗ (v)(−i)dΦ(v) = −i + dΦ0 (v)∗ (−i)dΦ(v) − idΦ0 (v) =: −i + Υω1 (v) and use the results of Lemma 1.65. To prove (ii), use Poincaré construction of the potential of ω1 which gives Z 1 Z 1 αω1 (v)u := h Eω1 (tv)tv, uidt = α0 (v)u + hWω1 (v), ui, Wω1 (v) = Υω1 (tv)tv dt , 0

0

where α0 is the potential for ω0 . In order to prove the analytic properties of Wω1 , note that R1 Wω1 (v) = 0 (H1 (tv) + H2 (tv))dt where H1 (v) := −i dΦ0 (v)v and H2 (v) := dΦ0 (v)∗ (−i)dΦ(v)v ≡ 0 ∗ 0 dΦ (v) (−iv + H1 (v)). Thus, by Lemma 1.65, one gets that kH1 kAw2 ≤ 2 Φ Aw2 ≤ 41 and w1 ,µ2 ρ w1 ,µρ

0 kH2 kAw2 ≤ 2 Φ Aw2 ≤ 41 . Thus the estimate on Wω1 follows. w1 ,µ3 ρ

w1 ,µ2 ρ

Remark 1.36. One has M αω1 − α0 = M αWω1 = hM Wω1 , ·i and kM Wω1 kAw2

w1 ,µ3 ρ

≤ kWω1 kAw2

.

w1 ,µ3 ρ

We are ready now for the first step. Lemma 1.37. There exists a map ϕˆ : BRw (µ5 ρ) → PRw such that (1 − ϕ) ˆ ∈ Aw w1 ,µ5 ρ and 1

1

k1 − ϕk ˆ Aw2

w1 ,µ5 ρ

Moreover ϕˆ satisfies the following properties: 26

≤ 25 1 .

2

(1.53)

(i) ϕˆ commutes with the rotations φθ , namely φθ ϕ(v) ˆ = ϕ(φ ˆ θ v) for every θ ∈ T . (ii) Denote ω ˆ 1 := ϕˆ∗ ω1 , then M ω ˆ 1 = ω0 . Proof. We apply the Darboux procedure described at the beginning of this section with Ω0 = ω0 ˆωt := (−i + t(M Eω + i)). Write equation and Ω1 = M ω1 . Then Ωt is represented by the operator E 1 1 (1.43), with f ≡ 0, in terms of the operators defining the symplectic forms, getting the equation ˆ t Yˆ t = −M Wω (see also Remark 1.36). This equation can be solved by inverting the operator E ω1 1 ˆωt by Neumann series: E 1 Yˆ t := −(−i + tM Υω1 )−1 M Wω1 . (1.54) 2 By the results of Lemma 1.35 and Remark 1.36, Yˆ t is of class Aw w1 ,µ4 ρ and fulfills

sup Yˆ t

≤ 2 kM Wω1 kAw2

2

Aw1

t∈[0,1]

w1 ,µ3 ρ

w ,µ4 ρ

≤ 24 1 .

(1.55)

1 1 By Lemma 1.66 the vector field Yˆ t generates a flow ϕˆt : BRw (µ5 ρ) → P w such that ϕˆt − 1 is of 2 class Aw w1 ,µ5 ρ and satisfies

t

ϕˆ − 1 w2 A

w1 ,µ5 ρ

≤ 2 sup Yˆ t

2

Aw1

t∈[0,1]

≤ 2 5 1 .

w ,µ4 ρ

Therefore the map ϕˆ ≡ ϕˆt |t=1 exists, satisfies the claimed estimate (1.53) and furthermore ϕˆ∗ M ω1 = ω0 . We prove now item (i). The claim follows if we show that the vector field Yˆ t commutes with ˆωt (v))−1 . By construcrotations. To this aim consider equation (1.54), and define Jˆωt 1 (v) = (E 1 ˆωt commutes with rotations (cf. Remark 1.34), namely ∀ θ0 ∈ T one has tion the operator E 1 ˆ t (v)u = E ˆ t (φθ0 (v))φθ0 u. Then it follows that φθ 0 E ω1 ω1 φθ0 Yˆ t (v) = −φθ0 Jˆωt 1 (v)M Wω1 (v) = −Jˆωt 1 (φθ0 (v))φθ0 M Wω1 (v) = −Jˆωt (φθ0 (v))M Wω (φθ0 (v)) = Yˆ t (φθ0 (v)). 1

1

This proves item (i). Item (ii) then follows from item (i) since, defining ω ˆ 1 = ϕˆ∗ ω1 , one has the ∗ ∗ chain of identities M ω ˆ 1 = M ϕˆ ω1 = ϕˆ M ω1 = ω0 . The analytic properties of the symplectic form ω ˆ 1 can be studied in the same way as in Lemma 1.35; we get therefore the following corollary: Corollary 1.38. Denote by Eωˆ 1 the symplectic operator describing ω ˆ 1 = ϕˆ∗ ω1 . Then 1 1 2 1 (i) Eωˆ 1 = −i + Υωˆ 1 , with Υωˆ 1 ∈ Nµ5 ρ (PRw , L(PRw , PRw )) and Υωˆ 1 5 ≤ 27 µρ . µ ρ

(ii) Define W (v) :=

R1 0

2

Υωˆ 1 (tv)tv dt, then W ∈ Aw w1 ,µ7 ρ and kW kAw2

w1 ,µ7 ρ

Furthermore the 1-form αW := hW, .i satisfies dαW = ω ˆ 1 − ω0 .

27

≤ 27 1 .

Finally we will need also some analytic and geometric properties of the map ˇ := ϕˆ−1 ◦ Ψ. Ψ

(1.56)

ˇ The functions {Ψ(v)} j≥1 forms a new set of coordinates in a suitable neighborhood of the origin whose properties are given by the following corollary: ˇ : B w1 (µ8 ρ) → P w1 , defined in (1.56), satisfies the following properCorollary 1.39. The map Ψ R R ties:

0 ˇ ˇ 0 := Ψ ˇ − 1 ∈ Aw21 8 with Ψ ˇ w2 (i) dΨ(0) = 1 and Ψ ≤ 28 1 . w ,µ ρ

A

w1 ,µ8 ρ

ˇ ˇ coincides with the foliation defined by Ψ. (ii) F (0) = Ψ(F), namely the foliation defined by Ψ 2 ˇ j }j≥1 pairwise commute with respect to the symplectic form ω0 . (iii) The functionals { 1 Ψ 2

Proof. By Lemma 1.65 the map ϕˆ is invertible in BRw (µ6 ρ) and ϕˆ−1 = 1 + g, with g ∈ Aw w1 ,µ6 ρ ˇ = 1+Ψ ˇ 0 where Ψ ˇ 0 = Ψ0 + g ◦ (1 + Ψ0 ). By Remark 1.29, and kgkAw2 ≤ 26 1 . Then Ψ w1 ,µ6 ρ

0

0 14 ˇ 0 ∈ Aw21 8 and moreover Ψ ˇ w2

Ψ w2 ≤ 6µ , thus Lemma 1.65 i) implies that Ψ ≤ 1 w ,µ ρ A A 1

2

w1 ,µ7 ρ

w1 ,µ8 ρ

6µ14 1 + 27 1 ≤ 28 1 . Item (ii) and (iii) follow from the fact that, by Lemma 1.37 (i), ϕˆ commutes with the rotations (see also the proof of Corollary 1.32). The second step consists in transforming ω ˆ 1 into the symplectic form ω0 while preserving the functions Il . In order to perform this transformation, we apply once more the Darboux procedure with Ω1 = ω ˆ 1 and Ω0 = ω0 . However, we require each leaf of the foliation to be invariant under the transformation. In practice, we look for a change of coordinates ϕ satisfying ϕ∗ Ω1 = Ω0 , Il (ϕ(v)) = Il (v),

(1.57) ∀l ≥ 1 .

(1.58)

In order to fulfill the second equation, we take advantage of the arbitrariness of f in equation (1.43). It turns out that if f satisfies the set of differential equations given by df (XI0l ) − (α1 − α0 )(XI0l ) = 0,

∀l ≥ 1

(1.59)

then equation (1.58) is satisfied (as it will be proved below). Here α1 is the potential form of ω ˆ 1 and is given by α1 := α0 + αW , where αW is defined in Corollary 1.38. However, (1.59) is essentially a system of equations for the potential of a 1-form on a torus, so there is a solvability condition. In Lemma 1.42 below we will prove that the system (1.59) has a solution if the following conditions are satisfied: d(α1 − α0 )|T F (0) = 0 ,

(1.60)

M (α1 − α0 )|T F (0) = 0 .

(1.61)

In order to show that these two conditions are fulfilled, we need a preliminary result. First, for v ∈ PRw fixed, define the symplectic orthogonal of Tv F (0) with respect to the form ω t := ω0 + t(ˆ ω1 − ω0 ) by n o (Tv F (0) )∠t := h ∈ PRw : ω t (v)(u, h) = 0 ∀u ∈ Tv F (0) . (1.62) 28

1

Lemma 1.40. For v ∈ BRw (µ5 ρ), one has Tv F (0) = (Tv F (0) )∠t . Proof. First of all we have that, since for any couple of functions F, G and any change of coordinates Φ, one has {F ◦ Φ, G ◦ Φ}Φ∗ ω0 = {F, G}ω0 ◦ Φ , it follows that

n o 2 2 {Il , Im }ω1 = |Ψl | , |Ψm |

=0,

∀l, m ≥ 1

ω0

and {Il , Im }ωb1 ◦ ϕˆ−1 = Il ◦ ϕˆ−1 , Im ◦ ϕˆ−1 ω1 but, by the property of invariance with respect to rotations of ϕ b (and therefore of ϕ b−1 ), Ij ◦ ϕ b−1 is a function of {Il }l≥1 only, and therefore the above quantity vanishes and one has ∀l, m 0 = {Il (v), Im (v)}ωˆ 1 = h∇Il (v), Jωˆ 1 (v)∇Im (v)i = hvl el , Jωˆ 1 (v)vm em i

∀ l, m ≥ 1.

(1.63)

Define Σv := span {vl el , l ≥ 1}. The identities (1.63) imply that Jωˆ 1 (v)(Σv ) ⊆ Σ⊥ v ≡ iΣv . By w1 5 Corollary 1.38 (i), Eωˆ 1 (v) is an isomorphism for v ∈ BR (µ ρ), so the same is true for its inverse Jωˆ 1 (v). Hence Jωˆ 1 (v)(Σv ) = iΣv and Σv = Eωˆ 1 (v)(iΣv ) and ω ˆ 1 (XI0l , XI0m ) = hEωˆ 1 (v)(ivl el ), ivm em i = 0,

∀ l, m ≥ 1.

(1.64)

Since ω t is a linear combination of ω0 and ω ˆ 1 , the previous formula implies that ω t (v)(XI0l , XI0m ) = 0 1 for every t ∈ [0, 1] and v ∈ BRw (µ5 ρ), hence Tv F (0) ⊆ (Tv F (0) )∠t . Now assume by contradiction that the inclusion is strict: then there exists u ∈ (Tv F (0) )∠t , kuk = 1, such that u ∈ / Tv F (0) . (0) (0) ⊥ Decompose u = u> + u⊥ with u> ∈ Tv F and u⊥ ∈ (Tv F ) . Due to the bilinearity of ω(v)t , we can always assume that u ≡ u⊥ . Then for every l ≥ 1

dIl (v)(−iu) = h∇Il (v), −iui = −iXI0l (v), −iu = XI0l (v), u = 0 ∀l ≥ 1 since XI0l (v) ∈ Tv F (0) . Hence iu ∈ Tv F (0) and therefore ω t (v)(−iu, u) = 0. Furthermore it holds that

ω t (0)(iu, u) = ω0 (−iu, u) = i2 u, u = −1. 1

It follows that for v ∈ BRw (µ5 ρ) one has ktM Υωˆ 1 (v)kL(P w1 ,P w1 ) ≤ 1/2, thus ω t (v)(iu, u) = −1 + R R htM Υωˆ 1 (v)iu, ui < 0, leading to a contradiction. We can now prove the following lemma: Lemma 1.41. The solvability conditions (1.60), (1.61) are fulfilled. Proof. Condition (1.60) follows by equation (1.64), since d(α1 − α0 )(XI0l , XI0m ) = ω ˆ 1 (XI0l , XI0m ) − ω0 (XI0l , XI0m ) = 0,

∀l, m ≥ 1.

We analyze now (1.61). We claim that in order to fulfill this condition, one must have that ω ˆ1 satisfies M ω ˆ 1 = ω0 , which holds by Lemma 1.37 (ii). Indeed, since 0 = Mω ˆ 1 − ω0 = M (ˆ ω1 − ω0 ) = M d(α1 − α0 ) = dM (α1 − α0 ), there exists a function g such that M (α1 − α0 ) = dg. But M dg = M (M (α1 − α0 )) = M (α1 − α0 ) = d g(φtl ) = dg(XI0l ) = M (α1 − dg, therefore g = M g, so g is invariant by rotations. Hence 0 = dt t=0 0 α0 )(XIl ), ∀l ≥ 1, thus also (1.61) is satisfied. 29

We show now that the system (1.59) can be solved and its solution has good analytic properties: Lemma 1.42. (Moser) If conditions (1.60) and (1.61) are fulfilled, then equation (1.59) has a solution f . Moreover, denoting hj := (α1 − α0 )(XI0j ), the solution f is given by the explicit formula f (v) =

∞ X

fj (v),

fj (v) = M1 · · · Mj−1 Lj hj

1 Lj g = 2π

Z

(1.65)

j=1

where

1

1

2π

tg(φtj )dt .

0 2

Finally f ∈ Nµ7 ρ (PRw , C), ∇f ∈ Nµ7 ρ (PRw , PRw ) and f 7 ≤ 210 1 µ7 ρ, ∇f 7 ≤ 211 1 . µ ρ µ ρ

(1.66)

Proof. Denote by θj the time along the flow generated by XI0j , then one has dg(XI0j ) = that the equations to be solved take the form ∂f = hj , ∂θj Clearly

∂ ∂θj Mj hj

∀j ≥ 1.

∂g ∂θj

, so

(1.67)

= 0, and by (1.60) it follows that ∂ ∂hj ∂hl ∂ Mj hj = Mj = Mj = Mj hl = 0, ∂θl ∂θl ∂θj ∂θj

∀l, j ≥ 1,

which shows that Mj hj is independent of all the θ’s, thus Mj hj = M hj . Furthermore, by (1.61) one has M hj = 0, ∀ j ≥ 1. Now, using that ∂θ∂ j Lj g = g − Mj g, one verifies that fj defined in (1.65) satisfies  0 if l < j  ∂fj M1 · · · Mj−1 hj if l = j =  ∂θl M1 · · · Mj−1 hl − M1 · · · Mj hl if l > j P where, for j = 1, we defined M1 · · · Mj−1 hl = hl . Thus the series f (v) := j≥1 fj (v), if convergent, satisfies (1.67). We prove now the convergence of the series for f and ∇f . First we define, for θ ∈ T , θ

Θθj := φθ11 · · · φj j

∀j ≥ 1 ,

then by (1.65) one has Z

fj (v) = θj hj (Θθj v) dθj , Tj Z θ j ∇fj (v) = Θ−θ j θj ∇hj (Θj v) dθ ,

(1.68) (1.69)

Tj

where T j is the j-dimensional torus and dθj =

dθ1 2π

···

dθj 2π .

Now, using that

hj (v) = hW (v), XI0j (v)i = Re(iWj (v)¯ vj ) 30

∀j ≥ 1

P∞ one gets that fj (|v|) ≤ 2π hj (|v|) ≤ 2π Wj (|v|)|vj |, therefore f (|v|) ≤ j=1 fj (|v|) ≤ 2π kW (|v|)kw1 kvkw1 and it follows that f µ7 ρ ≤ 2π |W |µ7 ρ µ7 ρ. This proves the convergence of the series defining f . Consider now the gradient of hj , whose k th component is given by ∂Wj (v) [∇hj (v)]k = Re i v¯j + δj,k Re (iWj (v)) . ∂vk Inserting the formula displayed above in (1.69) we get that ∇fj is the sum of two terms. We begin by the second one, which we denote by (∇fj )(2) . The k th component of (∇f )(2) := P estimating (2) is given by j (∇fj ) h

(∇f (v))

(2)

i k

  Z X (2)   = = (∇fj (v))

Tk

j

1

θ k Θ−θ k θk Re (iWk (Θk v)) dθ ,

(1.70)

k

thus, for any v ∈ BRw (µ7 ρ) one has (∇f (|v|))(2) (2) (∇f )

µ7 ρ

k

≤ 2π Wk (|v|) , and therefore

≤ 2π |W |µ7 ρ ≤ π28 1 . (1)

We come to the other term, which we denote by (∇fj ) . Its k th component is given by Z h i ∂Wj θ θj −θ (1) (∇fj (v)) = Θj θj Re i (Θj v)φj vj dθ . ∂vk k Tj Then ∇fj (|v|) ≤ 2π

∂Wj ∂vk (|v|)|vj | th

It follows that the k

(1.71)

= 2π[dW (|v|)]jk |vj |.

component of the function (∇f )(1) :=

(1) j (∇fj )

P

satisfies

  i h X X (∇f (|v|))(1) ≤  (∇fj (|v|))(1)  ≤ 2π [dW (|v|)]jk |vj | . k

Therefore (∇f )(1)

µ7 ρ ∗

j

≤ 2π kW kAw2

w1 ,µ7 ρ

k

j

≤ π28 1 . This is the step at which the control of the norm

of the modulus dW of dW ∗ is needed. Thus the claimed estimate for ∇f follows. We can finally apply the Darboux procedure in order to construct an analytic change of coordinates ϕ which satisfies (1.57) and (1.58). Lemma 1.43. There exists a map ϕ : BRw (µ9 ρ) → PRw which satisfies (1.57). Moreover ϕ − 1 ∈ 1 2 Nµ9 ρ (PRw , PRw ), ϕ − 1 = O(v 2 ) and ϕ − 1 9 ≤ 214 1 . (1.72) µ ρ 1

1

Proof. As anticipated just after Corollary 1.39, we apply the Darboux procedure with Ω0 = ω0 , Ω1 = ω ˆ 1 and f solution of (1.59). Then equation (1.43) takes the form Y t = (−i + tΥωˆ 1 )−1 (∇f − W ), 31

(1.73)

where Υωˆ 1 and W are defined in Corollary 1.38. By Lemma 1.42 and Corollary 1.38, the vector 1 2 field Y t is of class Nµ8 ρ (PRw , PRw ) and sup Y t µ8 ρ < 2(211 1 + 27 1 ) < 213 1 . t∈[0,1]

1

1

Thus Y t generates a flow ϕt : BRw (µ9 ρ) → PRw , defined for every t ∈ [0, 1], which satisfies (cf. Lemma 1.66) t ϕ − 1 9 ≤ 214 1 , ∀t ∈ [0, 1] . µ ρ

t

Thus the map ϕ := ϕ |t=1 exists and satisfies the claimed properties. We prove now that the map ϕ of Lemma 1.43 satisfies also equation (1.58). Lemma 1.44. Let f be as in (1.65) and ϕt be the flow map of the vector field Y t defined in (1.73). Then ∀ l ≥ 1 one has Il (ϕt (v)) = Il (v), for each t ∈ [0, 1]. Proof. The following chain of equivalences follows from Lemma 1.40 and the Darboux equation (1.43): d Il (ϕt (v)) = dIl (Y t (v)) ⇐⇒ Y t (v) ∈ Tv F (0) dt ⇐⇒ Y t (v) ∈ (Tv F (0) )∠t ⇐⇒ ωvt (Y t (v), XI0l (v)) = 0 , ∀l ≥ 1

Il (ϕt (v)) = Il (v) ⇐⇒ 0 =

⇐⇒ α1 (XI0l ) − α0 (XI0l ) = df (XI0l )

∀l ≥ 1 .

In turn the last property follows since f is a solution of (1.59). We can finally prove the quantitative version of the Kuksin-Perelman Theorem. Proof of Theorem 1.31. Consider the map ϕ of Lemma 1.43. Since dϕ(0) = 1, ϕ is invertible in 1 1 2 BRw (µ10 ρ) and ϕ−1 = 1 + g1 with g1 ∈ Nµ10 ρ (PRw , PRw ) and g1 µ10 ρ ≤ 2 ϕ − 1 µ9 ρ ≤ 215 1 (cf. Lemma 1.64). Define now e := ϕ−1 ◦ Ψ. ˇ Ψ e ∗ ω0 = ω0 , thus proving that Ψ e is symplectic. By equation (1.58) one has It’s easy to check that Ψ e ˇ e ˇ define the same foliation, which coincides Il (Ψ(v)) = Il (Ψ(v)) for every l ≥ 1, therefore Ψ and Ψ also with the o foliation defined by Ψ, c.f. Corollary 1.39. Similarly one proves that the functionals n 1 f pairwise commute with respect to the symplectic form ω0 . We have thus proved 2 Ψj (v) j≥1

item i) − iii) of Theorem 1.31. e e 0 := Ψ e −1 = Ψ ˇ 0 + g1 ◦ (1 + Ψ ˇ0 We prove now item iv). Clearly dΨ(0) = 1, and Ψ 0 ) is of 1 2 w w ˇ 11 ≤ class Nµ11 ρ (PR , PR ). Moreover, by Remark 1.29 and Corollary 1.39 (i), one has Ψ µ ρ 6 ˇ 0 6 9 11 0 10 ˇ 11 ≤ µ ρ and by Lemma 1.63 2µ Ψ 8 ≤ µ 2 1 ≤ µ ρ by condition (1.40). Thus 1 + Ψ µ ρ

e Ψ0

µ11 ρ

µ

0 ˇ 11 + g1 ◦ (1 + Ψ ˇ 0 ) ≤ Ψ µ ρ

µ11 ρ

ρ

0 ˇ 11 + g1 10 ≤ 28 1 + 215 1 ≤ 216 1 . ≤ Ψ µ ρ µ ρ

32

e 0 ∈ Aw21 12 . Since Ψ e ∗ ω0 = ω0 , one has dΨ(v) e ∗ (−i) Ψ(v) e We are left to prove that Ψ = −i, from w ,µ ρ 0 e which it follows that Ψ satisfies e 0 (v)∗ = i dΨ e 0 (v) dΨ

e 0 ∈ Aw21 12 with e 0 and therefore Ψ

Ψ

w ,µ ρ

3

−1

e 0 (v) 1 + dΨ

i

< 217 1 .

2

Aw1

w ,µ12 ρ

Toda lattice

3.1

Proof of Theorem 1.3 and Corollary 1.6.

We consider the Toda lattice with N particles and periodic boundary conditions on the positions q and momenta p: qj+N = qj , pj+N = pj , ∀ j ∈ Z. As anticipated in Section 1, we restrict to the invariant subspace characterized by (1.2). The phase space of the system is P s,σ , where s ≥ 0, σ ≥ 0 and it is defined in terms of the linear, complex, Birkhoff variables (ξ, η) (defined in (1.25)). PN −1 We endow the phase space with the symplectic form 2 Ω0 = −i k=1 dξk ∧ dηk . We will denote by PRs,σ the real subspace of P s,σ in which ηk = ξ¯k ∀1 ≤ k ≤ N − 1, endowed with the norm (1.7), and by BRs,σ (ρ) the ball in PRs,σ with center 0 and radius ρ > 0. The main step of the proof of Theorem 1.3 is the construction of the functions {Ψj }1≤j≤N −1 . This is based on a detailed analysis of the spectrum of the Jacobi matrix appearing in the Lax pair representation of the Toda lattice. So we start by recalling the elements of the theory needed for our development. Introduce the translated Flaschka coordinates [Fla74] by (b, a) = Θ(p, q),

1

(bj , aj ) := (−pj , e 2 (qj −qj+1 ) − 1).

(1.75)

The translation of the a variables by 1 is useful in order to keep the equilibrium point at (b, a) = (0, 0). Recall that the variables b, a are constrained by the conditions N −1 X

bj = 0,

j=0

N −1 Y

(1 + aj ) = 1 .

j=0

Introduce Fourier variables (ˆb, a ˆ) for the Flaschka coordinates by (1.3). In these variables Ek =

|ˆbk |2 + 4|ˆ ak |2 + O(ˆ a3 ), 2

1≤k ≤N −1 .

(1.76)

The Jacobi matrix whose spectrum forms a complete set of integrals of motions for the Toda lattice 2 so

that the Hamilton equations become ∂H ξ˙k = i , ∂ηk

η˙ k = −i

33

∂H , ∂ξk

(1.74)

is given by [vM76]      L(b, a) :=    

b0

1 + a0

0

1 + a0

b1

1 + a1

0 .. .

1 + a1 .. .

b2 .. .

1 + aN −1

...

0

 1 + aN −1  ..  .   . 0   1 + aN −2  bN −1

... .. . .. . .. . 1 + aN −2

(1.77)

It is useful to double the size of L(b, a), redefining 

Lb,a

         :=          

b0

1 + a0

1 + a0 .. .

b1 .. . .. . ...

0 0 0 .. . 1 + aN −1

... .. . .. . 1 + aN −2 0

...

...

0

0

... ...

1 + aN −2

0 .. .

bN −1 1 + aN −1

1 + aN −1 b0

... 1 + a0

0 .. .

1 + a0 .. . 0

0 .. .

0

0

1 + aN −1



0 .. .

         .         

b1 .. .

0 ... .. . .. .

1 + aN −2

..

1 + aN −2

bN −1

.

0 0 .. .

(1.78)

Consider the eigenvalues of Lb,a and order them in the non-decreasing sequence λ0 (b, a) < λ1 (b, a) ≤ λ2 (b, a) < . . . < λ2N −3 (b, a) ≤ λ2N −2 (b, a) < λ2N −1 (b, a) where one has that where the sign ≤ appears equality is possible, while it is impossible in the correspondence of a sign 0, independent of N , and an analytic map ∗ Ψ : B s,σ , Ω → P s,σ , (ξ, η) 7→ (φ(ξ, η), ψ(ξ, η)) 0 N2 such that: ¯ = ψk (ξ, ξ) ¯ ∀k. (Ψ1) Ψ is real for real sequences, namely φk (ξ, ξ) (Ψ2) For every 1 ≤ j ≤ N − 1, and for (φ, ψ) ∈ B s,σ N∗2 ∩ PRs,σ , one has 2 2 γj2 = N2 ω Nj |ψj | = N2 ω Nj |ϕj | . 34

(1.80)

(Ψ3) Ψ(0, 0) = (0, 0) and dΨ(0, 0) = 1. (Ψ4) There exist constants C1 , C2 > 0, independent of N , such that for every 0 < ≤ ∗ , the map Ψ0 := Ψ − 1 ∈ N/N 2 P s,σ , P s+1,σ and [dΨ0 ]∗ ∈ N/N 2 P s,σ , L(P s,σ , P s+1,σ ) . Furthermore one has 2 0 0 ∗ Ψ 2 ≤ C1 ; ] (1.81) [dΨ 2 ≤ C2 . /N N2 /N The main point is (Ψ4), in which the estimates of the domain of definition of the map Ψ holds uniformly in the limit N → ∞. We show now how Theorem 1.3 follows from Kuksin-Perelman Theorem 1.31. k 1/2 N −1 }k=1 and w2 := Proof of Theorem 1.3. Introduce the weights w1 := {N 3/2 [k]sN eσ[k]N ω N 1 k 1/2 N −1 σ[k]N {N 3/2 [k]s+1 }k=1 and consider the map Ψ of Theorem 1.45 as a map from P w in ω N N e 1 itself. Since for any (ξ, η) ∈ P w one has that k(ξ, η)kP w1 ≡ N 2 k(ξ, η)kP s,σ ,

(1.82)

it follows by scaling that there exists a constant C3 > 0, independent of N , such that

0

Ψ w2 ≤ C3 ρ2 . A w1 ,ρ

Thus, for any ρ ≤ ρ∗ ≡ min

2−34 C3 , ∗

, Ψ satisfies condition (1.40). Thus we can apply Theorem e defined on B w1 (aρ∗ ) 1.31 to the map Ψ, getting the existence of a symplectic real analytic map Ψ which satisfies i) − iv) of Theorem 1.31. e is invertible in B w1 (µaρ∗ ) and its inverse Φ satisfies Φ = 1 + Φ0 with By Lemma 1.65 the map Ψ 2 Φ0 ∈ A w w1 ,µaρ∗ . To get the statement of the theorem simply reexpress the map Φ in terms of real variables (x, y), (X, Y ) and denote such a map by ΦN . Remark 1.46. By the proof of Theorem 1.3 above one deduces the estimate

0

dΦ (φ, ψ)∗ s,σ s+1,σ ≤ Cs,σ Rs,σ , sup L(P ,P )

(1.83)

k(φ,ψ)kP s,σ ≤Rs,σ /N 2

for some Cs,σ > 0, independent of N . The rest of this subsection is devoted to the proof of Theorem 1.45. In the following it will be convenient to consider the variables (b, a) defined in (1.75) dropping PN −1 QN −1 the conditions j=0 bj = 0 and j=0 (1 + aj ) = 1. Equation (1.76) suggests to introduce on the variables b, a the norm 2

k(b, a)kC s,σ :=

N −1 1 X 2σ[k]N max(1, [k]2s |ˆbk |2 + 4|ˆ ak |2 N )e 2N

(1.84)

k=0

and to define the space CRs,σ := (b, a) ∈ RN × RN : k(b, a)kC s,σ < ∞ . 35

(1.85)

We will write C s,σ for the complexification of CRs,σ . In the following we will consider normally analytic map between the spaces P s,σ and C s,σ . We need to specify the basis of C s,σ that we will use to verify the property of being normally analytic. While it is quite hard to verify this property when the basis is general, it turns out that it is quite easy to verify it using the basis of complex exponentials defined in (1.3). Indeed the norm (1.84) is given in term of the Fourier variables. For the same reason, it will be convenient to express a map from C s,σ to P s,σ as a function of the Fourier variables ˆb, a ˆ. We prove now some analytic properties of the map Θ defined in (1.75). In the following we will denote by ΘΞ the map Θ expressed in the (ξ, η) variables. Proposition 1.47. The map ΘΞ satisfies the following properties: (Θ1) ΘΞ (0, 0) = (0, 0). Furthermore let dΘΞ (0, 0) be the linearization of ΘΞ at (ξ, η) = (0, 0). Then (B, A) = dΘΞ (0, 0)[(ξ, η)] iff 1 2ω

b0 = 0, B

bk = − B

b0 = 0, A

bk = −i$k A

k N

1/2

(ξk + ηN −k ), 1≤k ≤N −1 , −1/2 k 2ω N (ξk − ηN −k ), 1 ≤ k ≤ N − 1.

(1.86)

where $k := (1 − e−2iπk/N )/2, ∀ 1 ≤ k ≤ N − 1. Moreover for any s ≥ 0, σ ≥ 0 there exist constants CΘ1 , CΘ2 > 0, independent of N , such that

dΘΞ (0, 0) s,σ s,σ ≤ CΘ , 1 L(P ,C )

dΘΞ (0, 0)∗ s+2,σ s+1,σ ≤ CΘ2 . L(C ,P ) N

(1.87)

(Θ2) Let Θ0Ξ := ΘΞ − dΘΞ (0, 0). For any s ≥ 0, σ ≥ 0, there exist constants CΘ3 , CΘ4 , ∗ > 0, independent of N , such that the map Θ0Ξ ∈ N∗ /N 2 (P s,σ , C s+1,σ ) and the map [dΘ0Ξ ]∗ ∈ N∗ /N 2 (P s,σ , L(C s+2,σ , P s+1,σ )), and 0 ΘΞ

/N 2

[dΘ0Ξ ]∗

≡

sup k(ξ,η)kP s,σ ≤/N 2

/N 2

≡

0

ΘΞ (ξ, η)

≤

C s+1,σ

sup k(ξ,η)kP s,σ ≤/N 2

0

dΘΞ (ξ, η)∗

CΘ3 2 ; N2

L(C s+2,σ , P s+1,σ )

≤

CΘ4 . N2

(1.88)

The proof of the proposition is postponed in Appendix C. Note that the estimates (1.87) and (1.88) imply that there exists a constant CΘ5 > 0, independent of N , such that for any ρ ≤ N∗2 one has ΘΞ ∈ Nρ (P s,σ , C s,σ ) and ΘΞ ≤ CΘ5 ρ . (1.89) ρ

We start now the perturbative construction of the Birkhoff coordinates for the Toda lattice, which is based on the construction of the spectrum and of the eigenfunctions of Lb,a (defined in (1.78)) as a perturbation of the free operator L0 := Lb,a |(b,a)=(0,0) . More precisely we decompose Lb,a = L0 +Lp ,

36

where 

0

1

0

   L0 =    

1 0 .. .

0 1 .. .

1 0 .. .

... .. . ... .. .

1

...

...

1

  b0 1  ..   a0 .     0 0  , L = p     ..   . 1 aN −1 0

a0

0

b1 a1 .. .

a1 b2 .. .

... .. . ... .. .

...

...

aN −2

 aN −1  ..  .  0    aN −2  bN −1

(1.90)

and following the approach in [KP10, BGGK93, Kap91] we apply Kato perturbation theory [Kat66]. The next lemma characterizes completely the spectrum of L0 as an operator on C2N : Lemma 1.48. Consider L0 as an operator on C2N , then its eigenvalues and normalized eigenvectors are: eigenvalues λ00 = −2, λ02j−1 = λ02j = −2 cos λ02N −1 = 2,

eigenvectors k 1 f00 (k) = √2N (−1) jπ 1 1 √ e−iρj k , f2j,0 (k) = √2N eiρj k , 1≤j ≤N −1 N , f2j−1,0 (k) = 2N 1 f2N −1,0 (k) = √2N where 0 ≤ k ≤ 2N − 1 and ρj := 1 + Nj π. In particular the gaps of L0 are all closed. The proof is an easy computation and can be found in [HK08b]. Remark 1.49. For 0 ≤ j, k ≤ bN/2c one has λ02j − λ02k , λ02N −j − λ02N −k ≥ In particular if j 6= k then λ02j − λ02k ≥ 1/N 2 .

4|j 2 −k2 | . N2

We use now Kato perturbation theory of operators in order to introduce the main objects needed in the following and to give some preliminary estimates. For 1 ≤ j ≤ N − 1 let Ej (b, a) be the two-dimensional subspace spanned by the eigenvectors corresponding to the eigenvalues λ2j−1 (b, a) and λ2j (b, a) of Lb,a . Analogously, let E0 (b, a) (respectively EN (b, a)) be the one-dimensional subspace spanned by the eigenvector of λ0 (b, a) (respectively λ2N −1 (b, a)). Introduce the spectral projector on Ej (b, a) defined by I 1 −1 Pj (b, a) = − (Lb,a − λ) dλ, 0≤j≤N (1.91) 2πi Γj where, for 1 ≤ j ≤ N − 1, Γj is a closed path counter-clockwise oriented in C which encloses the eigenvalues λ2j−1 (b, a) and λ2j (b, a) and does not contain any other eigenvalue of Lb,a . Analogously, Γ0 (respectively ΓN ) encloses the eigenvalue λ0 (b, a) (respectively λ2N −1 (b, a)) and no other eigenvalue of Lb,a . Pj (b, a) maps C2N onto Ej (b, a) and, as we will prove, is well defined for (b, a) small enough. Pj (0, 0) will be denoted by Pj0 and its range Ej (0, 0), which will be denoted by Ej0 , is given by Im Pj0 = Ej0 , Ej0 = span hf2j,0 , f2j−1,0 i . Define also the transformation operators −1/2 2 Uj (b, a) = 1 − (Pj (b, a) − Pj0 ) Pj (b, a),

1 ≤ j ≤ N − 1.

(1.92)

Uj has the property of mapping isometrically Ej0 into the subspace Ej (b, a) spanned by the perturbed eigenvectors [Kat66]. Remark, however, that in general the image of an unperturbed eigenvector is not an eigenvector itself. We prove now some properties of the just defined objects. 37

Lemma 1.50. There exist a constant Cs,σ > 0, independent of N , such that the map (b, a) 7→ Lp (b, a) is analytic as a map from C s,σ to L C2N . Moreover kLp (b, a)kL(C2N ) ≤ Cs,σ k(b, a)kC s,σ .

(1.93)

Then by Kato theory one has the corollary Corollary 1.51. There exist constants Cs,σ , ∗ > 0, independent of N , such that the following holds true: s,σ ∗ (i) The spectrum of Lb,a is close to the spectrum of L0 ; in particular for any (b, a) ∈ B C N2 λ2j (b, a) − λ02j , λ2j−1 (b, a) − λ02j−1 ≤ Cs,σ k(b, a)k s,σ . (1.94) C (ii) One has that (b, a) 7→ Pj (b, a) is analytic as a map from B C s,σ ∗ (b, a) ∈ B C N 2 one has

s,σ

∗ N2

to L(C2N ). Moreover for

kPj (b, a) − Pj0 kL(C2N ) ≤ Cs,σ k(b, a)kC s,σ .

(1.95)

(iii) For each 1 ≤ j ≤ N − 1, the maps Uj , defined in (1.92), are well defined from B C L(C2N ) and satisfy the following algebraic properties:

s,σ

∗ N2

to

(U 1) Im Uj (b, a) = Ej (b, a); (U 2) for (b, a) real, one has Uj (b, a)f = Uj (b, a)f¯; (U 3) for (b, a) real and f ∈ Ej0 , one has kUj (b, a)f kC2N = kf kC2N . Finally the following analytic property holds: (U 4) One has that (b, a) 7→ Uj (b, a) is analytic as a map from B C s,σ ∗ for (b, a) ∈ B C N 2 one has

s,σ

∗ N2

to L(C2N ). Moreover

2

kUj (b, a) − Pj (b, a)kL(C2N ) ≤ Cs,σ k(b, a)kC s,σ .

(1.96)

The proofs of Lemma 1.50 and Corollary 1.51 can be found in Appendix D. s,σ ∗ For 1 ≤ j ≤ N − 1 and (b, a) ∈ B C N 2 define now the vectors f2j−1 (b, a) := Uj (b, a)f2j−1,0 ,

and

f2j (b, a) := Uj (b, a)f2j,0

which by property (U 1) belong to Ej (b, a). Define also the maps E −1/2 D zj (b, a) := N2 ω Nj Lb,a − λ02j f2j (b, a), f2j (b, a) , E −1/2 D wj (b, a) := N2 ω Nj Lb,a − λ02j−1 f2j−1 (b, a), f2j−1 (b, a)

(1.97)

(1.98)

P where hu, vi = uj vj is the Hermitian product in C2N . Finally denote z(b, a) = (z1 (b, a), . . . , zN −1 (b, a)) and w(b, a) = (w1 (b, a), . . . , wN −1 (b, a)), and let Z be the map (b, a) 7→ Z(b, a) := (z(b, a), w(b, a)). 38

(1.99)

The map Ψ of Theorem 1.45 will be constructed by expressing Z as a function of the linear Birkhoff coordinates ξ, η. The properties of the map Z are collected in the next lemma which constitutes the main technical step for the application of Kuksin-Perelman Theorem to the Toda lattice. s,σ ∗ Lemma 1.52. The map Z, defined by (1.99), is well defined for (b, a) ∈ B C N 2 . If b, a are real valued and fulfill k(b, a)kC s,σ ≤ N∗2 , then, for every 1 ≤ j ≤ N − 1, the following properties are also fulfilled: (Z1) zj (b, a) = wj (b, a); 2 (Z2) γj2 = N2 ω Nj |zj (b, a)| =

2 Nω

j N

2

|wj (b, a)| ;

(Z3) zj (0, 0) = wj (0, 0) = 0; moreover the linearizations of zj and wj at (b, a) = (0, 0) are given by −1/2 ˆj − 2ejiπ/N Aˆj , dzj (0, 0)[(B, A)] = 2ω Nj B (1.100) −1/2 ˆN −j − 2e−jiπ/N AˆN −j . B dwj (0, 0)[(B, A)] = 2ω Nj The map dZ(0, 0) = (dz(0, 0), dw(0, 0)) is in the class L(C s,σ , P s,σ ). Its adjoint dZ(0, 0)∗ is in the class L(P s,σ , C s+1,σ ). Finally there exist constants CZ1 , CZ2 > 0, independent of N , such that for any s ≥ 0 and σ ≥ 0 kdZ(0, 0)kL(C s,σ , P s,σ ) ≤ CZ1 ,

kdZ(0, 0)∗ kL(P s,σ , C s+2,σ ) ≤ CZ2 N 2 .

(1.101)

(Z4) For any s ≥ 0, σ ≥ 0, there exist constants CZ3 , CZ4 , ∗ > 0, independent of N , such that for every 0 < ≤ ∗ the map Z 0 := Z − dZ(0, 0) ∈ N/N 2 C s,σ , P s+1,σ and the map [dZ 0 ]∗ ∈ N/N 2 C s,σ , L(P s,σ , C s+2,σ ) . Moreover 2

0

Z (b, a) s+1,σ ≤ CZ3 , P N2 k(b,a)kCs,σ ≤/N 2

0

dZ (b, a)∗ s,σ s+2,σ ≤ CZ4 N . sup L(P ,C )

sup

(1.102)

k(b,a)kCs,σ ≤/N 2

The proof of the lemma is very technical, and is postponed in Appendix E. Remark 1.53. In the limit of infinitely many particles, the linearization dzj (0, 0)(b, a) at the different edges of the spectrum are given by ˆj − 2Aˆj B dzj (0, 0)(B, A) ≈ p 2ω(j/N )

if j/N 1

ˆj + 2Aˆj B dzj (0, 0)(B, A) ≈ p 2ω(j/N )

if 1 − j/N 1 .

(1.103) The existence of two different sequences is in agreement with the works [BKP13b, BKP13a], in which the spectrum of the Lax operator associated to the Toda lattice is approximated, up to a small error, by the spectrum of two Sturm-Liouville operators associated to two KdV equations. R R More explicitly, in [BKP13b] the following result is proved: take α, β ∈ C ∞ (T) such that T α = T β = 0, aj = 1 + N12 α(j/N ) and bj = N12 β(j/N ). Then the spectrum of the Lax matrix (1.78) with aj , bj as elements can be approximated at the two edges by the spectrum of the two Sturm-Liouville operators d2 ∞ (T). L = − dx 2 + (β ± 2α) on C 39

We are ready to define the map Ψ of Theorem 1.45: let Ψ : P s,σ → P s,σ ,

(ξ, η) 7→ (φ(ξ, η), ψ(ξ, η))

(1.104)

defined by Ψ = −Z ◦ ΘΞ ;

i. e.

φ = −z ◦ ΘΞ ,

ψ = −w ◦ ΘΞ .

(1.105)

We show now that Ψ satisfies the properties (Ψ1) − (Ψ4) claimed in Theorem 1.45. Proof of Theorem 1.45. Property (Ψ1) and (Ψ2) follows by (Z1) respectively (Z2). We prove now (Ψ3). By (Θ1) and (Z3) one has Ψ(0, 0) = (0, 0). In order to compute dΨ(0, 0) = (dφ(0, 0), dψ(0, 0)) note that dφ(0, 0) = −dz(0, 0) dΘΞ (0, 0) = −(dz(0, 0)F −1 ) ◦ (FdΘΞ (0, 0)) . ˆ A) ˆ = FdΘΞ (0, 0)(ξ, η). Then (1.100) and (1.86) imply that, for 1 ≤ j ≤ N − 1, Let (B, dφj (0, 0)(ξ, η) = − p

1

2ω(j/N )

ˆj − 2eiπj/N Aˆj B

r

1

=p 2ω(j/N ) where we used that 2eiπj/N $j = iω

! ω(j/N ) 2eiπj/N $j (ξj + ηN −j ) − i p (ξj − ηN −j ) ≡ ξj , 2 2ω(j/N )

j N

. One verifies analogously that dψj (0, 0)(ξ, η) = ηj .

We prove now property (Ψ4), which is a consequence of the fact that the space of normally analytic maps is closed by composition (see Lemma 1.63). Fix s ≥ 0 and σ ≥ 0. Let 0 < ≤ CΘ∗ , where 5

CΘ5 is the constant in (1.89). Since Z = dZ(0, 0) + Z 0 and ΘΞ = dΘΞ (0, 0) + Θ0Ξ , one gets that Ψ0 = −Z 0 ◦ ΘΞ − dZ(0, 0) ◦ Θ0Ξ .

(1.106)

Thus properties (Z3), (Θ2) and estimate (1.89) imply that there exists a constant C > 0, independent of N , such that 0 Ψ

/N 2

≡

2

0

Ψ (ξ, η) s+1,σ ≤ C , P N2 k(ξ,η)kP s,σ ≤/N 2

sup

which proves the first estimate of (Ψ4). We study now the adjoint map dΨ0 (ξ, η)∗ . Writing dΘΞ = dΘΞ (0, 0) + dΘ0Ξ one gets that dΨ0 (ξ, η)∗ = −dΘΞ (0, 0)∗ dZ 0 (ΘΞ (ξ, η))∗ − dΘ0Ξ (ξ, η)∗ dZ 0 (ΘΞ (ξ, η))∗ − dΘ0Ξ (ξ, η)∗ dZ(0, 0)∗ = I + II + III. We estimate each term in the expression displayed above. In the following, if A ∈ Nρ (P s,σ , L(P s,σ , P s+1,σ )), we denote by |A|ρ ≡ sup kA(ξ, η)kL(P s,σ , P s+1,σ ) . k(ξ,η)kP s,σ ≤/N 2

We begin by estimating I: |I|/N 2 ≤

CΘ2 N

0

dZ (ΘΞ (ξ, η))∗ s,σ s+2,σ ≤ CΘ2 CZ4 CΘ5 N ≤ C, L(P ,C ) N k(ξ,η)kP s,σ ≤/N 2 sup

40

where in the first inequality we used the second estimate of (1.87) and in the second inequality we used the second estimate in (1.102). Now we study II: |II|/N 2 ≤

2

0

CΘ4

dZ (ΘΞ (ξ, η))∗ s,σ s+2,σ ≤ CΘ4 CZ4 CΘ5 N ≤ C , sup L(P ,C ) N 2 k(ξ,η)kP s,σ ≤/N 2 N2 N

where we used the second estimate in (1.88) and again (Z4). Finally, using again (Θ2) and the second estimate of (1.101), one has |III|/N 2 ≤

CΘ4 CΘ4 kdZ(0, 0)∗ kL(P s,σ , C s+2,σ ) ≤ CZ2 N 2 ≤ C . 2 N N2

Collecting the estimates above one gets [dΨ0 ]∗ 2 ≡ sup /N

k(ξ,η)kP s,σ ≤/N 2

0

dΨ (ξ, η)∗ s,σ s+1,σ ≤ 3C, L(P ,P )

and (Ψ4) follows. 0 Proof of Corollary 1.6. Provided 0 < R < Rs,σ is small enough, one has that w0 := Φ−1 N (v0 ) fulfills

kw0 kP s,σ ≤

R (1 + CR) , N2

and, denoting by w(t) the solution in Birkhoff coordinates, one has kw0 kP s,σ = kw(t)kP s,σ . Thus, 0 is small enough one has provided 0 < R < Rs,σ kv(t)kP s,σ = kΦN (w(t))kP s,σ ≤

R (1 + C 0 R) N2

which implies the thesis.

3.2

Proof of Theorem 1.7

The proof is based on the construction of the first terms of the Taylor expansion of ΦN through Birkhoff normal form. To this end we work with the complex variables (ξ, η) (defined in (1.25)) and will eventually restrict to the real subspace PRs,σ . Remark 1.54. Consider the Taylor expansion of ΦN at the origin, one has ΦN = 1 + QΦN + O(k(ξ, η)kP s,σ ) , 3

then QΦN is a bounded quadratic polynomial. Furthermore, since ΦN is canonical, QΦN is a Hamiltonian vector field, i.e. there exists a cubic complex valued polynomial χΦN s.t. QΦn is the Hamiltonian vector field of χΦN . We need a preliminary result about a uniqueness property of the transformation introducing Birkhoff coordinates (called below Birkhoff map).

41

Lemma 1.55. Let ΦN and ΨN be Birkhoff maps for HT oda , analytic in some neighborhood of the origin; assume that dΦN (0, 0) ≡ dΨN (0, 0) = 1 and denote by χΦN and χΨN the Hamiltonian functions corresponding to QΦN and QΨN respectively, then one has {H0 ; χΦN − χΨN } = 0 ,

(1.107)

where H0 is defined in (1.6). Proof. By a standard computation of the Taylor expansion one has HT oda ◦ ΦN = H0 + {H0 , χΦN } + H1 + h.o.t. where H1 is the function H1 (q) =

N −1 X j=0

(qj − qj+1 )3 6

Since ΦN is a Birkhoff map, the function HT oda ◦ ΦN is in Birkhoff normal form so in particular its Taylor expansion contains only terms of even degree. Thus the cubic terms in the expansion above must vanish: {H0 , χΦN } + H1 = 0. The same argument holds also for the map ΨN , thus the thesis follows. Remark 1.56. Writing as usual X

χΦN (ξ, η) =

χK,L ξ K η L ,

|K|+|L|=3

one gets that, since X

{H0 , χΦN } = −

iω · (K − L)χK,L ξ K η L ,

|K|+|L|=3

eq. (1.107) implies that, if for some K, L one has ω ·(K −L) 6= 0, then χK,L is unique and coincides HK,L with an obvious definition of HK,L . with iω·(K−L) Lemma 1.57. In terms of the variables (ξ, η) one has  H1 (ξ, η) =

X  k1 +k2 +k3 √ 1 √ √  √ (−1) N ωk1 ωk2 ωk3 (ξk1 ξk2 ξk3 + ηk1 ηk2 ηk3 )  12 2N k1 +k2 +k3 =0 mod N 1≤k1 ,k2 ,k3 ,≤N −1

 X

+3

(−1)

k1 +k2 −k3 N

k1 +k2 −k3 =0 mod N 1≤k1 ,k2 ,k3 ≤N −1

√

 √ √ ωk1 ωk2 ωk3 (ξk1 ξk2 ηk3 + ηk1 ηk2 ξk3 ) 

Proof. First remark that N −1 N −1 2πijk 2πijk 2πik iπk 1 X 1 X qj − qj+1 = √ qˆk 1 − e− N e− N = √ iωk e− N qˆk e− N , N k=0 N k=0

42

so that N −1 1 X i3 (qj − qj+1 )3 = 6 j=0 6N 3/2

iπ

X

ωk1 qˆk1 ωk2 qˆk2 ωk3 qˆk3 e− N (k1 +k2 +k3 ) 3

i 6N 1/2

e

2πij N (k1 +k2 +k3 )

j=0

k1 ,k2 ,k3

=

N −1 X

X

(−1)

k1 +k2 +k3 N

ωk1 qˆk1 ωk2 qˆk2 ωk3 qˆk3 .

k1 +k2 +k3 =0 mod N

Substituting ωk qˆk =

√

ωk

ξk − ηN −k √ i 2

and reorganizing the terms one gets the thesis. Lemma 1.58. For any s ≥ 0, σ ≥ 0, there exists C > 0 s.t. one has

φ 2

Q N (¯ v ) P s,σ ≥ CN 2 k¯ v kP s,σ ,

(1.108)

where v¯ = ((ξ1 , 0, 0, ..., 0), (ξ¯1 , 0, 0, ..., 0)) ∈ PRs,σ . Proof. In this proof, for clarity we denote η1 := ξ¯1 , and similarly for the other variables. We are v )]ξ2 of QΦN (¯ v ) and exploit the inequality going to compute the ξ2 component [QΦN (¯ 1/2

Φ

1 s σ2 1/2 1 ΦN 2s eσ2 ω2 ∂χΦN

Q N (¯

v ) P s,σ ≥ √ 2 e ω2 √ Q (¯ v ) ξ2 = √ (¯ v ) ; ∂η2 2 N 2N

(1.109)

the only monomials in χΦN contributing to such a quantity are quadratic in (ξ1 , η1 ) and linear in η2 , but due to the selection rule k1 ± k2 ± k3 = lN with a plus for the ξ’s and a minus for the η’s ¯ L ¯ K ¯ := (2, 0, ..., 0), and the only monomial contributing to the r.h.s. of (1.109) is χK, ¯ L ¯ ξ η with K ¯ L = (0, 1, 0, 0, ..., 0). Since π 2π 2π 3 1 ω · (K − L) = 2ω1 − ω2 = 4 sin − 2 sin = 3 +O 6= 0 , (1.110) N N N N5 such a coefficient is uniquely defined and, for the χΦN corresponding to any Birkhoff map, one has 1/2

1 ω1 ω2 χK, . ¯ L ¯ = √ 4 2N i(2ω1 − ω2 )

(1.111)

Inserting in (1.109) one has that its r.h.s. is equal to 1/2

2s eσ2 ω2 √ 2N

ω2 C 00 ω1 ω2 2 2 2 2 χK, |ξ1 | = C 0 k¯ v kP s,σ ≥ CN 2 k¯ v kP s,σ , ¯ L ¯ |ξ1 | = N |2ω1 − ω2 | |2ω1 − ω2 |

where C, C 0 and C 00 are numerical constants independent of N and we used the expansions of ω1 , ω2 in 1/N as well as equation (1.110). Proof of Theorem 1.7. The thesis immediately follows taking k¯ v kP s,σ = R/N α and imposing the inequality (1.11). 43

Proof of Corollary 1.9. By Cauchy inequality and assumption (1.12) QΦN fulfills

Φ R0 N 2α 2

Q N (¯ v ) P s,σ ≤ α0 2 k¯ v kP s,σ . N R

(1.112)

Comparing this inequality with (1.108), one gets R0 2α−α0 N ≥ C 00 N 2 , R2 which in particular implies the thesis.

4

FPU packet of modes: proofs.

In this section we prove the results stated in the subsection 1.2 about the persistence of the metastable packet in the FPU system. To clarify the procedure, we distinguish here between the (ξ, η) variables and the variables (p, q). Thus, we denote by T : (ξ, η) → (p, q) the change of coordinates of the phase space introducing the linear Birkhoff variables (ξ, η) defined in (1.25). Furthermore it is useful to use for the (p, q) variables the following norms 2

kqks,σ :=

N −1 1 X 2σ[k]N max(1, [k]2s |ˆ qk |2 , N)e N

(1.113)

k=0

and

k(p, q)kP s,σ := T −1 (p, q) P s,σ .

(1.114)

Lemma 1.59. Fix s ≥ 1, σ ≥ 0, then there exist constants C1 , C2 > 0, independent of N , such that for all (ξ, η) ∈ P s,σ and ∀l ≥ 2 one has kXHl ◦T (ξ, η)kP s,σ ≤ kXHl ◦T (ξ, η)kP s−1,σ

C1l l+1 k(ξ, η)kP s,σ , (l + 1)! C2l l+1 ≤ k(ξ, η)kP s,σ . N (l + 1)!

(1.115) (1.116)

Proof. Define the difference operators by S± : {qj }0≤j≤N −1 7→ {qj − qj±1 }0≤j≤N −1 ,

where qN ≡ q0 ,

(1.117)

and the operator [S+ (q)]l by n o l [S+ (q)] := (qj − qj+1 )l , j

so that XHl ◦T (ξ, η) =

1 l T −1 S− [S+ (T (ξ, η))] , 0 . (l + 1)!

(1.118)

By Lemma 1.70 and Remark 1.72 in Appendix B, there exists a constant Cs,σ > 0, independent of N , such that for every integer n ≥ 1

l+1 l+1 l+1 l+1

[S± (q)]l+1 ≤ Cs,σ kS± (q)ks,σ ≤ Cs,σ k(ξ, η)kP s,σ , (1.119) s,σ 44

where for the last inequality we have identified the

q) with the corresponding (ξ, η) vector.

couple (0, Then the thesis follows just remarking that T −1 (q, 0) P s,σ = kqks,σ , and that S− is bounded as an operator from P s,σ to itself, while one has k(S− (q), 0)kP s−1,σ ≤

C kqks,σ . N

Introducing the Birkhoff coordinates and using the standard formulae for the pull back of vector fields3 one has the following Corollary 1.60. Fix s ≥ 1 and σ ≥ 0, then there exist constants Rs,σ , C1 , C2 > 0, independent of N , such that for all w ≡ (φ, ψ) ∈ B s,σ (Rs,σ /N 2 ) one has kXHl ◦T ◦ΦN (w)kP s,σ ≤ kXHl ◦T ◦ΦN (w)kP s−1,σ

C1l l+1 kwkP s,σ , (l + 1)! C2l l+1 kwkP s,σ . ≤ N (l + 1)!

(1.120) (1.121)

Remark 1.61. Write ˜ F P U ≡ HF P U ◦ T ◦ Φ N = H ˜ T oda + H ˜P , H

(1.122)

where ˜ T oda := HT oda ◦ T ◦ ΦN , H

˜ P := (β − 1)H2 ◦ T ◦ ΦN + H (3) ◦ T ◦ ΦN , H

˜ P fulfills the following estimates then, provided R is small enough the vector field of H i h

4 3

X ˜ (w) , ≤ C |β − 1| kwk s,σ + C kwkP s,σ s,σ P HP P h i

C 3 4

X ˜ (w) ≤ |β − 1| kwk , s,σ + C kwkP s,σ s−1,σ P HP P N

(1.123)

(1.124) (1.125)

for all w ∈ B s,σ (R/N 2 ). ¯ In the following we denote by v(t) ≡ (ξ(t), ξ(t)) the solution of the FPU model in the original Cartesian coordinates (we restrict to the real subspace). We denote by w(t) := Φ−1 N (v(t)) the same solution in Birkhoff coordinates. 0 , T, C2 > 0 such that v0 ∈ BRs,σ NR2 Lemma 1.62. Fix s ≥ 2 and σ ≥ 0. Then there exist Rs,σ 4R 0 for with R ≤ Rs,σ implies v(t) ∈ BRs,σ N 2 |t| ≤ 3 Namely

T . R2 µ4 [|β − 1| + C2 Rµ2 ]

[Φ∗N X](x) = dΦ−1 N (ΦN (x))X(ΦN (x))

which gives the vector field of the transformed Hamiltonian due to the fact that ΦN is canonical

45

(1.126)

0 Proof. First consider w0 := Φ−1 (provided Rs,σ is small enough) one has N (v0 ) and remark that n o 2 s,σ 2R ˜ w0 ∈ BR N 2 . Denote by M (w) := kwkP s,σ . Since M, HT oda ≡ 0, one has R

M (w(t)) = M (w0 ) +

Z tn o ˜ P (w(s))ds . M; H

(1.127)

0

¯ (t) := sup|s|≤t M (w(s)), one has Denoting M Z t n o ˜ P (w(s)) ds ¯ M (w(t)) ≤ M (w0 ) + M; H 0 Z t 4 5 ≤ M (w0 ) + C kw(s)kP s,σ |β − 1| + C kw(s)kP s,σ ds R R 0 Z t ¯ (t)1/2 ds ¯ (t)2 |β − 1| + C M ≤ M (w0 ) + CM 0 ¯ (t)2 |β − 1| + C M ¯ (t)1/2 , ≤ M (w0 ) + |t|C M

(1.128)

(1.129)

n o ˜ P := dM X ˜ and where, in order to prove the second inequality we used M ; H HP kdM (w)kL(P s,σ ,C) ≤ C kwkP s,σ , which follows from an explicit computation. Taking t as in the statement of theLemma we have ¯ (t) ≤ 9M (w(0))/4, which implies w(t) ∈ B s,σ 3R2 from which the that (1.128)-(1.129) ensures M R N thesis immediately follows. Proof of Theorem 1.16. Inequality (1.21) is a direct consequence of Lemma 1.62. To prove inequality ˜ P } = xk ∂ H˜ P − yk ∂ H˜ P . Thus (1.22) remark that I˙k = {Ik , H ∂yk ∂xk N −1 ˜P ˜ P ∂H 1 X 2s−2 2σ[k]N ∂H k ˜ [k]N e ω N xk − yk {Ik , HP } = ∂yk N ∂xk k=1 k=1  2 2 1/2 !1/2  N −1 N −1 ∂H ∂H X ˜ ˜ 1 1 X 2s−2 2σ[k]N P P  2σ[k]N k k   [k]N e ω N (yk2 + x2k ) [k]2s−2 e ω + ≤ N N ∂yk ∂xk N N

N −1 1 X 2s−2 2σ[k]N [k]N e ω N

k N

k=1

k=1

i

C h 5 4 |β − 1| kwkP s,σ + C kwkP s,σ , ≤ 2 kwkP s−1,σ XH˜ P (w) P s−1,σ ≤ R N R where in the last inequality we used (1.125). Using that |Ik (w(t)) − Ik (w(0))| ≤ one gets N −1 1 X 2s−2 2σ[k]N [k]N e ω N

k N

|Ik (w(t)) − Ik (w(0))| ≤

k=1

which, using w(t) ∈ BRs,σ

3R N2

h i |t|C 4 5 sup |β − 1| kw(s)kP s,σ + C kw(s)kP s,σ , N |s|≤t

immediately implies the thesis.

46

R t ˜ P }(w(s)) ds, ,H 0 {Ik

A

Properties of normally analytic maps 1

2

2

In this section we study the properties of the space Nρ (P w , P w ) and Aw w1 ,ρ defined in section 2, with weights w1 ≤ w2 . In particular, we consider the operations on germs defined in [KP10] and perform quantitative estimates. 1

2

Lemma 1.63. Let w1 ≤ w2 ≤ w3 be weights. Let G ∈ Nρ (P w , P w ) with |G|ρ ≤ σ and F ∈ 2

3

1

3

Nσ (P w , P w ). Then F ◦ G ∈ Nρ (P w , P w ) and |F ◦ G|ρ ≤ |F |σ . Proof. Exploiting the obvious inequality F ◦ G(|v|) ≤ F ◦ G(|v|)(cf [KP10]), one has |F ◦ G|ρ ≡

sup v∈B w1 (ρ)

kF ◦ G(|v|)kw3 ≤

1

sup v∈B w1 (ρ)

kF (G(|v|))kw3 ≤

sup u∈B w2 (σ)

kF (|u|)kw3 ≡ |F |σ .

2

Lemma 1.64. Let F ∈ Nρ (P w , P w ), F = O(v 2 ) and |F |ρ ≤ ρ/e. Then the map w1

w1

1 + F is

w2

invertible in B (µρ), µ as in (1.50). Moreover there exists G ∈ Nµρ (P , P ), G = O(v 2 ), such that (1 + F )−1 = 1 − G, and |F |ρ . (1.130) |G|µρ ≤ 8 P n n Proof. We look for G in the form G = n≥2 G , with the homogeneous polynomial G to be determined at every order n. Note that the equation defining G can be given in the form F (v − G(v)) = G(v), which can be recasted in a recursive way giving the formula Gn (v) =

n X

F˜ r Gk1 (v), · · · , Gkr (v) ,

X

∀n ≥ 2 .

(1.131)

r=2 k1 +···+kr =n

P r r In the formula above k1 , . . . , kr ∈ N, and we write F = r≥2 F , where F is a homogeneous polynomial of degree r and F˜ r is its associated multilinear P map (see (1.29)). Moreover we write G1 (v) := v. We show now that the formal series G = n≥2 Gn with Gn defined by (1.131) is 1

normally analytic in B w (µρ). Note that Gn (|v|) ≤

n X

X

F˜ r Gk1 (|v|), . . . , Gkr (|v|) .

(1.132)

r=2 k1 +···+kr =n

In order to prove that the series constant A > 0 such that

P

n≥2

1

Gn is convergent in B w (µρ), we prove that there exists a

kGn (|v|)kw2 ≤

|F |ρ 8Sn2

n

An kvkw1 ,

∀n ≥ 2.

The proof is by induction on n. We will use in the following the chain of inequalities

r

˜ ∀r ≥ 1 ,

F ≤ er kF r k ≤ er |F |ρ /ρr

47

(1.133)

see [Muj86]. For n = 2, by (1.131) it follows that G2 (v) = F˜ 2 (v, v). Since

2

|F |ρ

2 2

G (|v|) 2 ≤

F˜ 2 kvkw1 ≤ e2 2 kvkw1 , w ρ 1/2

it follows that (1.133) holds for n = 2 with A = e(32S) . We prove now the inductive step ρ n−1 n. Assume therefore that (1.133) holds up to order n − 1. Then one has kGn (|v|)kw2 ≤ ≤

≤

n X

˜ r k1

F G (|v|) w2 · · · Gkr (|v|) w2

X

r=2 k1 +···+kr =n r n X X |F |ρ |F |ρ n An kvkw1 er r r r 2 ρ 8 S k1 · · · kr2 r=2 k1 +···+kr =n r ∞ X |F |ρ n |F |ρ n e |F |ρ n n A kvkw1 ≤ A kvkw1 2 2 4Sn 2ρ 8Sn r=2

where in the first inequality we used the fact that w1 ≤ w2 , in the second the inductive assumption and in the last we used the hypothesis |F |ρ ≤ ρ/e. Finally to pass from the second to the third line we used the following inequality, proved in Lemma (1.67) below: X

n2

k1 +···+kr =n

1 ≤ (4S)r−1 , k12 · · · kr2

n≥1.

(1.134)

Hence, choosing µρ = 1/A = ρ/e(32S)1/2 one proves (1.130). 2

Now it is easy to prove the following lemma, giving closedness of the class Aw w1 ,ρ under different operations. Lemma 1.65. Let w1 ≤ w2 be weights and let µ be as in (1.50). Then the following holds true: 2

2

w i) Let F ∈ Aw w1 ,ρ and G ∈ Aw1 ,µρ with kGkAw2

w1 ,µρ

1 one gets X

n2

k1 +···+kr =n

1 = 2 k1 · · · kr2

X k1 +j=n

n2 k12 j 2

X k2 +···+kr =j

X n2 j2 ≤ (4S)r−2 k22 · · · kr2 k12 j 2 k1 +j=k

by the induction assumption. Now it is enough to note that X k1 +j=n

n2 = k12 j 2

X k1 +j=n

n−1 n−1 X1 X 1 1 n2 ≤ 2 + ≤ 4 ≤ 4S. k12 (n − k1 )2 k12 (n − k1 )2 k12 k1 =1

50

k1 =1

B

Discrete Fourier Transform

In this section we collect some well-known properties of the discrete Fourier transform (DFT). For u ∈ CN , N ∈ N, the DFT of u is the vector u ˆ ∈ CN whose k th component is defined by N −1 1 X u ˆk = √ uj e2πijk/N , N j=0

∀ 0 ≤ k ≤ N − 1.

(1.142)

When the DFT is considered as a map, it will be denoted by F, i.e. F : u 7→ u ˆ. For any s ≥ 0 and σ ≥ 0 we endow CN with the norm k·ks,σ defined in (1.113). Such a space will be denoted by Cs,σ . Remark 1.68. Let j be an integer such that 0 ≤ j ≤ N − 1. Then ( √ N −1 2N −1 X X ˆl , 2 Nu 0 if j 6= 0 i2πjk/N iπkj/N e = and uk e = N if j = 0 0 k=0 k=0

j even, j = 2l j odd (1.143)

Remark 1.69. Fix s > 12 and σ ≥ 0. Then there exists a constant Cs,σ > 0, independent of N , such that for every u ∈ CN the following estimate holds: sup 0≤j≤N −1

|uj | ≤ Cs,σ kuks,σ .

For u, v ∈ CN , we denote by u · v the component-wise product of u and v, namely the vector whose j th component is given by the product of the j th components of u and v: (u · v)j := uj vj ,

0≤j ≤N −1 .

(1.144)

We denote by u ∗ v the convolution product of u and v, a vector whose j th component is defined by (u ∗ v)j :=

N −1 X

uk vj−k ,

0≤j ≤N −1 ,

(1.145)

k=0

where in the summation above u and v are extended periodically defining vk+lN ≡ vk for l ∈ Z. The DFT maps the component-wise product in convolution: Lemma 1.70. For s > the following holds: (i) ud ·v =

√1 N

1 2

and σ ≥ 0 there exists a constant Cs,σ > 0, independent of N , such that

u ˆ ∗ vˆ;

(ii) ku · vks,σ ≤ Cs,σ kuks,σ kvks,σ ; (iii) the map X : u 7→ u2 , has bounded modulus w.r.t. 2 Cs,σ kuks,σ .

51

the exponentials, and kX(u)ks,σ ≤

Proof. Item (i) is standard and the details of the proof are omitted. We prove now item (ii). To begin, note that, by periodicity, one has 2

kuks,σ =

1 X 2 [k]2s e2σ|k| |ˆ uk | , N 0 k∈KN

where the set 0 KN := {k ∈ Z : −(N − 1)/2 ≤ k ≤ (N − 1)/2} ∪ {bN/2c},

(1.146)

while [k] := max(1, |k mod N |). By item (i), one has that 2

ku · vks,σ

2 −1 NX X 1 1 X \ u ˆl vˆk−l . [k]2s e2σ|k| [k]2s e2σ|k| |(u · v)k |2 = 2 = N N 0 0

(1.147)

l=0

k∈KN

k∈KN

Introduce now the quantities γk,l :=

[l]s

[k]s eσ|k| · σ|l| σ|k−l| . s [k − l] e e

2s +[l]2s ) e2σ(|k−l|+|l|) 2 ≤ 4s [l]12s + For s > 12 and σ ≥ 0, it holds that γk,l ≤ 4s ([k−l] [k−l]2s [l]2s e2σ|l| e2σ|k−l| which it follows that there exists a constant Cs,σ > 0, independent of N , such that N −1 X

sup 0≤k≤N −1

1 [k−l]2s

2 2 γk,l ≤ Cs,σ .

, from

(1.148)

l=0

By Cauchy-Schwartz one has [k]s eσ|k|

N −1 X

|ˆ ul | |ˆ vk−l | =

N −1 X

γk,l [l]s eσ|l| |ˆ ul | [k − l]s eσ|k−l| |ˆ vk−l |

l=0

l=0

≤

N −1 X

!1/2 2 γk,l

l=0

N −1 X

!1/2 2s 2σ|l|

[l] e

2

2s 2σ|k−l|

|ˆ ul | [k − l] e

|ˆ vk−l |

2

.

l=0

Inserting the inequality above in (1.147), one has

ku · vks,σ

Cs,σ ≤ N

N −1 X

!1/2 2s 2σ|l|

[l] e

|ˆ ul |

2

l=0

N −1 X

!1/2 2s 2σ|k−l|

[k − l] e

2

|ˆ vk−l |

k=0

≤ Cs,σ kuks,σ kvks,σ . P b := FXF −1 . By item (i) one has X b : {ˆ ˆl u ˆj−l }j∈Z . We prove now item (iii). Consider X uj }j∈Z 7→ { √1N l u b b Thus X ≡ X and the claim follows. Remark 1.71. Let S± be the difference operators defined in (1.117). Let ω ˆ ± be the vectors whose k th components are given by ω ˆ ±,k := 1 − e∓2πik/N . Then the following holds: 52

(i) the map Sb± := FS± F −1 is a multiplication by the vector ω ˆ ± : Sb± : u ˆ 7→ ω ˆ± · u ˆ. N −1 k u|, where ω ≡ {ω N u) ≤ ω · |ˆ }k=1 (ii) Sb± (ˆ is the vector of the linear frequencies. Remark 1.72. Consider

q = q(ξ, η) as a function of the linear Birkhoff variables defined in (1.25).

Then one has S± (q) ≤ k(ξ, η)kP s,σ . s,σ

C

Proof of Proposition 1.47

We prove now property (Θ1). Let T : (ξ, η) 7→ (p, q) be the map introducing linear Birkhoff coordinated. Explicitly (p, q) = T (ξ, η) iff (ˆ p0 , qˆ0 ) = (0, 0) and r  1 1 k  (ˆ pk , qˆk ) =  (ξk + ηN −k ), q ω N (ξk − ηN −k ) , 1 ≤ k ≤ N − 1 . 2 i 2ω k N

Then ΘΞ ≡ Θ ◦ T and in particulardΘΞ (0, 0) = dΘ(0, 0)T . Using the formula above and the fact that dΘ(0, 0)(P, Q) = −P, 21 S + (Q) , where S+ is defined in (1.117), one obtains easily formula (1.86). The estimate of dΘΞ (0, 0) L(P s,σ , C s,σ ) is trivial, and is omitted.

We prove now the estimate for dΘΞ (0, 0)∗ s+1,σ s,σ . Using the explicit formula (1.86), L(C

,P

one computes that (ξ, η) = dΘΞ (0, 0)∗ (B, A) iff  q q $k k ˆ ˆk , − 1 ω (ξk , ηk ) = − 12 ω N Bk + q A 2 k i 2ω N

)

 k N

$k

ˆN −k − q B i 2ω

k N

ˆ  AN −k

for 1 ≤ k ≤ N − 1. Thus there exist constants C, CΘ2 > 0, independent of N , such that

dΘΞ (0, 0)∗ (B, A) s,σ ≤ C P

N −1 1 X 2s 2σ[k]N [k]N e ω N

k 2 N

!1/2 bk |2 + |A bk |2 ) (|B

≤

k=1

where we used that ω

k 2 ≤ N

π [k]2N N2

CΘ2 k(B, A)kC s+1,σ , N

2

. Thus the second of (1.87) is proved.

We prove now property (Θ2). Denote by Θb the map p 7→ −p and by Θa the map q 7→ exp 12 S+ (q) − 1. Then (b, a) = Θ(p, q) ≡ (Θb (p), Θa (q)). Introduce on CN the norm k·ks,σ defined 2 2 2 in (1.113). Then kΘ(p, q)kC s,σ ≡ kΘb (p)ks,σ + kΘa (q)ks,σ . The analyticity of p 7→ Θb (p) is obvious. Consider now the map q 7→ Θa (q). Expand Θa in Taylor series with center at the origin to get Θa (q) =

X

Θra (q),

Θra (q) :=

r≥1

1 (S+ (q))r , r! 2r

∀r ≥ 1.

(1.149)

Consider q as a function of the linear Birkhoff variables ξ, η. Then Lemma 1.70 and Remark 1.72 imply that for any s ≥ 0, σ ≥ 0

r

r

r r ≤ C1r S+ (q) ≤ C2r k(ξ, η)kP s+1,σ ≤ C3r N r k(ξ, η)kP s,σ , ∀r ≥ 2, (1.150)

Θa (q) s+1,σ

s+1,σ

53

1 C3

where C1 , C2 , C3 > 0 are positive constants independent of N . Therefore for < sup k(ξ,η)kP s,σ ≤/N 2

0

ΘΞ (ξ, η)

C s+1,σ

≤

X

sup

r≥2 k(ξ,η)kP s,σ ≤/N

2

r

ΘΞ (ξ, η)

C s+1,σ

≤

X

C3r N r

r≥2

one has

r 2C32 2 ≤ . N 2r N2

This proves the first estimate in (Θ2). We show now that for any s ≥ 0, σ ≥ 0 one has [dΘ0Ξ ]∗ ∈ N/N 2 (P s,σ , L(C s+2,σ , P s+1,σ )). Note that dΘΞ (ξ, η)∗ = T ∗ dΘ(T (ξ, η))∗ . Using the explicit expression of T , one verifies that (ξ, η) = T ∗ (P, Q) iff   q q 1 1 k b 1 k b b b  q (ξk , ηk ) =  12 ω N Pk + q (1.151) Qk , QN −k , 2 ω N PN −k − k k i 2ω N i 2ω N for 1 ≤ k ≤ N − 1. Thus one has that for any s ≥ 0, σ ≥ 0 kT ∗ (0, Q)kP s,σ ≤ kQks,σ .

(1.152) 0, S+ (q)r−1 · S+ (Q) , ∀r ≥ 2, from

1 Using (1.149) one verifies that dΘr (p, q)(P, Q) = (r−1)! 2r which it follows that r−1 1 dΘr (p, q)∗ (B, A) = S (q) · S (A) , 0, + − (r − 1)! 2r

∀r ≥ 2 .

Thus, using estimate (1.152), there exists a constant C4 > 0, independent of N , such that

r−1

r

r−1 ≤ C4r N r−2 k(ξ, η)kP s,σ k(B, A)kC s+2,σ .

dΘΞ (ξ, η)∗ (B, A) s+1,σ ≤ C4r S+ (q(ξ, η))

S− (A) P

s+1,σ

s+1,σ

Then there exists C5 , 0 > 0, independent of N , such that ∀ 0 < ≤ 0

X

r

0

≤ sup sup

dΘΞ (ξ, η)∗

dΘΞ (ξ, η)∗ k(ξ,η)kP s,σ ≤/N 2

L(C s+2,σ , P s+1,σ )

r≥2 k(ξ,η)kP s,σ ≤/N

≤

X

C4r N r−2

r≥2

D

L(C s+2,σ , P s+1,σ )

2

C5 r−1 ≤ 2. N N 2(r−1)

Proof of Lemma 1.50 and Corollary 1.51

Proof of Lemma 1.50. Since the map (b, a) 7→ Lp (b, a) is linear, it is enough to prove that it is continuous from C s,σ to L(C2N ). In particular we will prove that kLp kL(C2N ) ≤ sup |bj | + 2 sup |aj | . (1.153) 0≤j≤N −1

This estimate, together with Lemma 1.69, proves (1.93). D + A+ + A− , where D is the diagonal part of Lp and A±    0 a0    ..    . 0 A+ =   , A− =    0 aN −1  aN −1 0 54

j

In order to prove (1.153), write Lp = are defined by  0 aN −1  a0 0  . ..  . 0 aN −2

0

To estimate the norms of D, A+ and A− is enough to observe that for every x ∈ C2N one has 2 2 2N −1 X

± 2 2 2 2 2

kDxkC2N := |bj xj | ≤ sup |bj | kxkC2N , A x C2N ≤ sup |aj | kxkC2N , 0≤j≤N −1

j=0

0≤j≤N −1

where k·kC2N is the standard euclidean norm on C2N . Thus (1.153) follows. Proof of Corollary 1.51. Item (i) follows by standard perturbation theory, and are the details omitted. We prove now item (ii). Let Γj be the circle defined by Γj := λ ∈ C : λ02j − λ = 2N1 2 , counter-clockwise oriented. By item (i), for any k(b, a)kC s,σ ≤ N∗ , λ2j (b, a) and λ2j−1 (b,a) are inside −1

the ball enclosed by Γj . Write Lb,a − λ = L0 − λ + Lp = (L0 − λ) 1 + (L0 − λ) ! ∞ n X −1 −1 −1 (Lb,a − λ) = − (L0 − λ) Lp (L0 − λ)

Lp ; its inverse

(1.154)

n=0

−1 is well defined as a Neumann operator when (L0 − λ) Lp

L(C2N )

< 1. Since L0 − λ is diagonal-

izable with {(λ0j − λ)}0≤j≤2N −1 as eigenvalues, the norm of its inverse is bounded by the inverse of the smallest eigenvalue:

1

−1 2 sup (L0 − λ) 2N ≤ sup (1.155) λ0 − λ < 2N L(C ) λ∈Γ λ∈Γ j

k

j

0≤k≤2N −1

where the last estimates is due to the form of Γj . Therefore for 0 < ≤ ∗ and k(b, a)kC s,σ < one gets, using (1.93),

−1 −1

(L0 − λ) Lp 2N ≤ kLp kL(C2N ) (L0 − λ) 2N ≤ Cs,σ k(b, a)kC s,σ 2N 2 < 2Cs,σ ∗ , L(C

L(C

)

N2

)

which proves the convergence of the Neumann series (1.154) for ∗ ≤ 2C1s,σ . Substituting (1.154) in (1.91) we get, for 1 ≤ j ≤ N − 1, ! I ∞ n X 1 −1 −1 − (L0 − λ) Lp Pj (b, a) = Pj0 − (L0 − λ) dλ. 2πi Γj n=1

(1.156)

s,σ Since the series inside the integral is absolutely and uniformly convergent for (b, a) ∈ B C , 2 N C s,σ 2N (b, a) 7→ Pj (b, a) is analytic as a map from B ). Estimate (1.95) follows easily N 2 to L(C from (1.156). We prove now item (iii). Properties (U 1) − (U 3) are standard [Kat66]. The analyticity of the map (b, a) 7→ Uj (b, a) follows from item (ii). Indeed, in order for Uj (b, a) to be defined as a Neumann series one needs kPj (b, a) − Pj0 kL(C2N ) < 1, which follows from (1.95). Estimate (1.96) follows by expanding (1.92) in power series of Pj (b, a) − Pj0 .

E

Proof of Proposition 1.52

Denote by D : CN −1 → CN −1 the diagonal operator D : {ξj }1≤j≤N −1 7→ {Dj ξj }1≤j≤N −1 ,

where Dj := 55

2 Nω

j N

−1/2

.

(1.157)

Proof of properties (Z1) − (Z3). Property (Z1) follows from formula (1.98), since 4 :

zj (b, a) = Dj Lb,a − λ02j Uj f2j,0 , Uj f2j,0 = Dj Uj f2j,0 , Lb,a − λ02j Uj f2j,0 =

= Dj Uj f2j−1,0 , Lb,a − λ02j Uj f2j−1,0 = Dj Lb,a − λ02j f2j−1 , f2j−1 = wj (b, a). We prove now (Z2). Using Lemma 1.51 (iv) and the fact that f2j,0 = f2j−1,0 , decompose f2j,0 and f2j in real and imaginary part: f2j,0 = ej,0 + ihj,0 ,

f2j = ej + ihj

f2j−1,0 = ej,0 − ihj,0 ,

f2j−1 = ej − ihj ,

where ej,0 := Re f2j,0 ,

hj,0 := Im f2j,0 ,

and

ej := Re f2j = Uj ej,0 ,

hj := Im f2j = Uj hj,0 .

The vectors {ej , hj } form a real orthogonal basis for Ej (b, a). Let Mj (b, a) be the matrix of the selfadjoint operator Lb,a − λ02j E (b,a) with respect to this basis: j αj σj Mj (b, a) = . σj βj The eigenvalues of Mj are obviously λ2j − λ02j and λ2j−1 − λ02j , hence Tr Mj = αj + βj = λ2j − λ02j + λ2j−1 − λ02j , Det Mj = αj βj − σj2 = λ2j − λ02j λ2j−1 − λ02j . Now observe that Lb,a − λ02j (ej + ihj ), (ej − ihj ) =

= Dj Lb,a − λ02j ej , ej − Dj Lb,a − λ02j hj , hj + 2iDj Lb,a − λ02j ej , hj = −1/2 (αj − βj + i2σj ). = N2 ω Nj

zj (b, a) = Dj

Finally one computes 2

2

(λ2j − λ2j−1 ) = (Tr Mj )2 − 4Det Mj = (αj + βj ) − 4αj βj + 4σj2 2

2

2

= (αj − βj ) + 4σj2 = (Re zj ) + (Im zj ) =

2 Nω

j N

|zj (b, a)|2 .

We prove now (Z3). The first order terms of zj and wj in (b, a) are given by

dzj (0, 0)(b, a) = Dj Lp f2j,0 , f2j,0 , dwj (0, 0)(b, a) = Dj Lp f2j−1,0 , f2j−1,0 ,

1 ≤ j ≤ N −1 .

Using the explicit formula for f2j,0 in Lemma 1.48, one computes

2N −1 1 X Lp f2j,0 , f2j,0 = bl ei2ρj l + al−1 ei2ρj (l−1) eiρj + al ei2ρj l eiρj 2N l=0

=

1 2N

2N −1 X

bl ei2πjl/N + al−1 ei2π(l−1)j/N eiρj + al ei2πlj/N eiρj

l=0

1 ˆ 1 ˆ =√ bj + 2eiρj a ˆj = √ bj − 2eiπj/N a ˆj . N N 4 to

simplify the notation, we write fj ≡ fj (b, a) and Uj ≡ Uj (b, a)

56

(1.158)

The formula for dzj (0, 0)(b, a) immediately follows. The one for dwj (0, 0)(b, a) is proved in the same way and the details are omitted. The estimate (1.101) for dZ(0, 0) follows immediately. We estimate now the norm of dZ(0, 0)∗ . b0 = A b0 = 0 and for 1 ≤ k ≤ N − 1 One checks that (B, A) = dZ(0, 0)∗ (ξ, η) iff B   1 2 iπk/N bk , A bk ) =  q (B ξk + eiπ(N −k)/N ηN −k ) . (ξk + ηN −k ), q (e k k 2ω N 2ω N Thus there exist constants C, C 0 , CZ > 0, independent of N , such that N −1 C 0 X 2s 2σ[k]N ω [k]N e N

[k]4N 2 2 2 2 4 |ξk | + |ηk | ≤ CZ N k(ξ, η)kP s,σ k 2 ω k=1 N 2 k where in the last inequality we used that [k]4N /ω N ≤ C 00 N 4 for some constant C 00 > 0 independent of N . Thus the second of (1.101) is proved. 2

kdZ(0, 0)∗ (ξ, η)kC s+2,σ ≤

k N

Proof of property (Z4). We will prove that Z is normally analytic. Recall that, as mentioned in the discussion before Proposition 1.47, the map Z is said to be normally analytic if Zˇ := ZF is normally analytic. With an abuse of notations, we omit the “check” from Z. We begin by expanding the components of Z, denoted by Zj (b, a) := (zj (b, a), wj (b, a)), in Taylor series with center at (b, a) = (0, 0). The first two terms of the expansions are given by −1 (1 − Pj0 ) Lp f2j,0 , f2j,0 i + O((b, a)3 ), zj (b, a) = Dj hLp f2j,0 , f2j,0 i + Dj hLp L0 − λ02j −1 wj (b, a) = Dj hLp f2j−1,0 , f2j−1,0 i + Dj hLp L0 − λ02j (1 − Pj0 ) Lp f2j−1,0 , f2j−1,0 i + O((b, a)3 ). (1.159) To perform the Taylor expansion at every order it is convenient to proceed in the following way. Write zj (b, a) = zj,1 (b, a) + zj,2 (b, a) and wj (b, a) = wj,1 (b, a) + wj,2 (b, a) where E E D D zj,1 (b, a) = Dj L0 − λ02j f2j (b, a), f2j (b, a) , zj,2 (b, a) = Dj Lp f2j (b, a), f2j (b, a) , (1.160) while wj,1 (b, a) and wj,2 (b, a) are defined as in (1.160), but with f2j−1 (b, a) replacingP f2j (b, a). n Expand zj,ς (b, a), ς = 1, 2, in Taylor series with center at (b, a) = (0, 0): zj,ς (b, a) = n≥1 zj,ς (b, a), n with zj,ς a homogeneous polynomial of degree n in b, a. We write an analogous expansion for wj,ς (b, a). Therefore one has n n n n Zjn (b, a) := (zjn (b, a), wjn (b, a)) ≡ zj,1 (b, a) + zj,2 (b, a), wj,1 (b, a) + wj,2 (b, a) . n In order to write explicitly zj,ς (b, a) as a function of b and a, one needs to expand the vectors f2j (b, a) and f2j−1 (b, a) in Taylor series of b, a. Rewrite (1.92), (1.97) as −1/2 2 f2j (b, a) = Uj (b, a)f2j,0 = 1 − (Pj (b, a) − Pj0 ) 1 + (Pj (b, a) − Pj0 ) f2j,0

and expand the r.h.s. above in power series of Pj (b, a) − Pj0 , getting: f2j (b, a) =

∞ X

m

cm (Pj (b, a) − Pj0 ) f2j,0 ,

f2j−1 (b, a) =

m=0

∞ X

m

cm (Pj (b, a) − Pj0 ) f2j−1,0 ,

m=0

(1.161) 57

1+x where the cm ’s are the coefficients of the Taylor series of the function φ(x) = (1−x 2 )1/2 . Note that −1/2 1 1 1 k −1/2 c2k+1 = c2k ≡ (−1) , where k := − 2 (− 2 −1) · · · (− 2 −k +1) is the product of k negative k terms, thus (−1)k −1/2 ≥ 0, ∀k ≥ 0, and therefore cm ≥ 0, ∀m. k s,σ

By Corollary 1.51 (see also formula (1.156)) one has, in the ball B C (∗ /N 2 ), I ∞ i X −1 (−1)n T n (b, a, λ) (L0 − λ) dλ Pj (b, a) − Pj0 = 2π n=1 Γj

(1.162)

where the Γj ’s are defined as in equation (1.91), and T (b, a, λ) := (L0 − λ) Substituting (1.162) in (1.161) we get that X X X f2j (b, a) = f2j,0 + cm n≥1 1≤m≤n α f2j,m (b, a) m

i 2π

α=(α1 ,...,αm

−1

Lp .

α f2j,m (b, a),

)∈Nm , |α|=n

:=

(−1)|α|

I

I ...

−1

T α1 (b, a, λ1 ) (L0 − λ1 )

−1

. . . T αm (b, a, λm ) (L0 − λm )

f2j,0 dλ1 . . . dλm .

Γj

Γj

(1.163) An analogous expansion holds for f2j−1 (b, a), with f2j−1,0 substituting f2j,0 in the integral formula above. In order to write explicitly the expression inside the integral, one needs to compute the iterated terms T n (b, a, λ)f2j,0 and T n (b, a, λ)f2j−1,0 . The computation turns out to be simpler if we express Lp f2j,0 in the basis of the eigenvectors of L0 . To simplify the notations we relabel the eigenvectors of L0 in the following way: g0 := f00 ,

gN := f2N −1,0 ,

gj := f2j,0 ,

g−j := f2j−1,0 ,

for 1 ≤ j ≤ N − 1

and the eigenvalues of L0 as ˆ 0 := λ0 , λ 0

ˆ N := λ0 λ 2N −1 ,

ˆ j := λ0 , λ 2j

ˆ −j := λ0 , λ 2j−1

for 1 ≤ j ≤ N − 1.

ˆj = λ ˆ −j For every 1 ≤ j ≤ N − 1 one has that gj = g−j , formally, one can also write gj+2N = gj , λ ˆ ˆ ˆ and λj+2N = λj , as one verifies using the explicit expressions of the gj ’s and λj ’s. In this notation, ˆ ±j , one has (L0 − λ)−1 g±j = g±j /(λ ˆ ±j − λ). With a computation analogous to the one in for λ 6= λ (1.158) (using also the second formula in (1.143)), one verifies that the projection of Lp gj on the vector gk is given by 1 ˆ hLp gj , gk i = √ b j−k − 2 cos kπ a ˆ δ(j−k; even ) , (1.164) j−k N 2 2 N where δ(j−k; even ) = 1 if j − k is an even integer, and equals 0 otherwise. Formula (1.164) implies that Lp gj is supported only on the vectors gk whose index k satisfies k = j − 2l for some integer l. Therefore we can write X xlj 1 ˆ gj−2l , xlj := hLp gj , gj−2l i = √ bl − 2 cos (j−2l)π a ˆl , T (b, a, λ)gj = N ˆ j−2l − λ N λ l∈K 0 N

(1.165) 58

0 where KN is the set of indexes defined in (1.146). Note that xlj ≤

√2 N

|ˆbl | + |ˆ al | uniformly in

j, and xl+N = xlj . Iterating (1.165) one gets j −1

T n (b, a, λ) (L0 − λ)

X

gj =

0 i1 ,··· ,in ∈KN

n 2 . . . xij−2i xij1 xij−2i 1 −···−2in−1 1 gj−2i1 −···−2in . Qn ˆ ˆ P (λj − λ) l=1 λj−2 l im − λ m=1

More generally, for a vector α = (α1 , . . . , αm ) ∈ N T

αm

(b, a, λm ) (L0 − λm )

−1

···T

α1

0 i1 ,...,in ∈KN

ˆ j − λ1 ) (λ

with |α| = n and λ1 , · · · , λm ∈ Γj , one has −1

(b, a, λ1 ) (L0 − λ1 ) 2 xij1 xij−2i 1

X

=

m

Qn ˆ P l=1 λj−2 l

− µl

m=1 im

gj =

n . . . xij−2i 1 −···−2in−1

Q

m−1 l=1

ˆ Pα1 +···+αl λ − λl+1 j−2 i h=1

gj−2i1 −···−2in

h

(1.166) where µl = λ1 for 1 ≤ l ≤ α1 ,

and µl = λk for

k−1 X

αh + 1 ≤ l ≤

h=1

k X

αh ,

2≤k≤m.

(1.167)

h=1

n n To obtain the explicit expression of zj,ς and wj,ς , ς = 1, 2, in terms of the Fourier variables ˆb, a ˆ, we n substitute (1.166) in (1.163) and the obtained result in (1.160). By (1.163), zj,1 is a sum of terms D E α β of the form L0 − λ02j f2j,p , f2j,p over (p, α, β) ∈ N2 × Np1 × Np2 with |p| = p1 + p2 ≤ n and 1 2

|α| + |β| = n. For |α| = r, |β| = n − r one gets I I D E i |p| β ir i1 i2 n α ˆ L0 − λj f2j,p1 , f2j,p2 = (−1) ... κp,α,β j,1 (i)xj xj−2i1 . . . xj−2i1 −···−2ir−1 × 2π Γj Γj

ir+1 in in−1 × xj xj−2in . . . xj−2in −···−2ir+2 gj−2i1 −···−2ir , gj−2ir+1 −···−2in dλ1 . . . dλ|p| , (1.168) where, writing i = (i1 , · · · , in ), ˆj ˆ j−2 Pr i − λ λ m=1 m Q × κp,α,β j,1 (i) := Q r p1 −1 ˆ ˆ j − λ1 ) ˆ Pl P α +···+α (λ λ − µ λ − λ 1 l l+1 l i j−2 m=1 im l=1 l=1 j−2 h=1 h (1.169) 1 Q × p2 −1 ˆ ˆ j − λp +1 ) Qn ˆ j−2 Pn i − µ Pβ1 +···+βl (λ λ ˜l λ − λl+1 1

l=r+1

l=1

m=l m

j−2

h=1

ih

and the µ ˜l ’s are defined as in (1.167), but with the E multi-index β replacing α. Similarly, the term D β n α f over (p, α, β) ∈ N2 × Np1 × Np2 with |p| ≤ n and zj,2 is a sum of terms of the form Lp f2j,p , 2j,p 1 D E 2 β α |α| + |β| = n − 1. The term Lp f2j,p , f2j,p has an expression similar to (1.168), and for |α| = r 1 2 and |β| = n − 1 − r the kernel κp,α,β j,2 (i) is given by κp,α,β j,2 (i) :=

ˆ j − λ1 ) Qr ˆ P (λ l=1 λj−2 l

m=1 im

×

1 Q

− µl

p1 −1 l=1

ˆ Pα1 +···+αl λ − λl+1 j−2 i h=1

×

h

.

1 ˆ j − λp +1 ) (λ 1

Qn

l=r+2

ˆ j−2 Pn i − µ λ ˜l m=l m 59

Q

p2 −1 l=1

ˆ Pβ1 +···+βl λ − λl+1 j−2 i h=1

h

(1.170)

Using the explicit form of the eigenvectors {gk }−(N −1)≤k≤N (see Lemma 1.48), one verifies that

gj−2i1 −···−2ir , gj−2ir+1 −···−2in = δ j,

n X

! im

,

gN −j−2i1 −···−2ir , gN −j−2ir+1 −···−2in = δ −j,

m=1

m=1

This is used to simplify the last term in (1.168). Moreover, using j = ˆj = λ ˆ −j , one gets that λ ˆ j−2i = λ ˆ P λ n j−2 n−1

m=1 im

,

Pn

m=1 im

and the identity

ˆ j−2i −2i −···−2i ˆ j−2 Pr i . λ =λ n n−1 r+1 m=1 m

...,

(1.171)

Recalling the definition of the coefficients xlj (formula (1.165)), we can write, for ς = 1, 2, n ˆ zj,ς (b, a ˆ) =

1 N n/2

j N

2 Nω

−1/2

X

n Kj,ς (i, ι) ui1 ,ι1 . . . uin ,ιn

(1.172)

(i,ι)∈∆n

where the set 0 ∆n := (i, ι) ∈ Zn × Nn : il ∈ KN ,

ιl ∈ {1, 2},

∀1 ≤ l ≤ n ,

the variables u = (ui1 ,ι1 , · · · , uin ,ιn ) are defined by uir ,1 := ˆbir ,

uir ,2 := a îr ,

n the kernels Kj,ς (i, ι) are defined for (i, ι) ∈ ∆n by

Y

n n ˜ j,ς Kj,ς (i, ι) := K (i)

−2 cos

(j−2ii −···−2il )π N

ιl −1

,

(1.173)

{1≤l≤n}

X

n ˜ j,ς K (i) =

X

cp1 cp2

p,α,β Sj,ς (i)

(1.174)

(α,β)∈Np1 ×Np2 |α|=r, |β|=s

r+s=n−(ς−1) p=(p1 ,p2 )∈N2 , |p|≤n

and finally p,α,β Sj,ς (i)

= δ j,

n X m=1

! im

i 2π

|p|

n X

(−1)n

I

I ...

Γj

κp,α,β (i) dλ1 . . . dλ|p| . j,ς

(1.175)

Γj

n n An analogous expansion holds also for wj,1 and wj,2 . n We need now to get estimates of the kernels Kj,ς , which will follow from estimates on the denominators of κp,α,β . j,ς n o 1/2 hji hN −ji , where hji = 1 + |j|2 . Lemma 1.73. Let µ ∈ Γj := λ ∈ C : λ − λ02j = min 2N 2 , 2N 2

Then there exists a constant R > 0, independent of N , such that for every −(N − 1) ≤ k ≤ N one has ( Rhj − kihj + ki/N 2 , if 0 ≤ |j| ≤ bN/2c ˆ (1.176) λk − µ ≥ 2 Rhj − kih(N − j) + (N − k)i/N , if bN/2c + 1 ≤ |j| ≤ N

60

! im

.

ˆ j and λ ˆ k are in the low half of Proof. Consider first the situation in which both the eigenvalues λ the spectrum, namely 0 ≤ |j|, |k| ≤ bN/2c. In this case one has ˆk − λ ˆ j | ≡ |λ0 − λ0 | = 2 cos |k|π − cos |j|π = 2 cos |λ 2|k| 2|j| N N

kπ N

− cos

jπ N

4|j 2 − k 2 | ≥ . N2

Therefore, for k 6= j, there exists a positive constant R1 such that for ∀µ ∈ Γj hji 4|j 2 − k 2 | hji hj − ki hj + ki ˆ ˆ ˆ ≥ − ≥ R1 , λk − µ ≥ λ k − λj − 2N 2 N2 2N 2 N2

(1.177)

where we used the inequality hji ≤ 2 hj − ki hj + ki, which holds since j, k are integers. If k = j, ˆ k − µ| = hji/2N 2 . then the claimed estimate follows trivially since |λ ˆ j is in the low half of the spectrum, while λ ˆ k is in the high half, i.e. Consider now the case when λ ˆ j and λ ˆ k is 0 ≤ |j| ≤ bN/2c, while bN/2c < |k| ≤ N . In this case the distance of the eigenvalues λ 1 2 of order N , therefore the estimate (1.176) holds as well. More precisely, using cos x ≥ 1 − π x for 0 ≤ x ≤ π/2, one has ˆk − λ ˆ j | = |λ0 − λ0 | = 2 cos (N −|k|)π + cos |λ 2|k| 2|j| N

jπ N

4(|k| − |j|) hj − ki hj + ki ≥ , ≥ N N2

where the last inequality holds since hli /N ≤ 4, ∀|l| ≤ 2N . The inequality above implies that ˆ ˆk − λ ˆ j | − hji ≥ hj − ki hj + ki − hji ≥ R2 hj − ki hj + ki , λk − µ ≥ |λ 2N 2 N2 2N 2 N2

(1.178)

for some R2 > 0. Thus the first of (1.176) is proved. The proof of the second inequality of (1.176) follows by symmetry and is omitted. n We can now estimate the kernels Kj,ς defined in (1.173). n Lemma 1.74. There exists a constant R > 0, independent of N , such that Kj,ς (i, ι), ς = 1, 2, satisfy, for every n ≥ 2 and 1 ≤ j ≤ bN/2c, the estimates ! n X n 1 n 2(n−1) Kj,ς (i, ι) ≤ R N D E DP E, il Q δ j, n−1 Pl l i i − j k k l=1 l=1 k=1 k=1 ! (1.179) n X n 1 KN −j,ς (i, ι) ≤ Rn N 2(n−1) δ −j, D E DP E. il Q n−1 Pl l l=1 l=1 k=1 ik k=1 ik − j

Proof. We start by estimating κp,α,β (i), defined in (1.169) and (1.170). For every −(N −1) ≤ k ≤ N j,ς ˆ ˆ , therefore and µ ∈ Γj one has λk − µ ≥ λj − µ ≥ min hji2 , hN −ji 2 2N

2N

pY pY 1 −1 2 −1 ˆ ˆ ˆ ˆ λj−2 Pα1 +···+αl i − λl+1 (λj − λp1 +1 ) λj−2 Pβ1 +···+βl i − λl+1 (λj − λ1 ) h h h=1 h=1 l=1 l=1 |p| hji hN − ji ≥ min , . 2N 2 2N 2 61

Let now 1 ≤ j ≤ bN/2c. By Lemma 1.73, formula (1.171) and the inequality

ˆj | ˆ j−2 Pr −λ |λ m=1 im P ˆ |λj−2 r i −µ|

≤2

m=1 m

(which is used to estimate just p,α,β κj,ς (i) ≤ h

min

where

κp,α,β j,1 (i)),

hji hN −ji 2N 2 , 2N 2

it follows that, for ς = 1, 2, 2 aj (i1 , · · · , in−1 ) ≤h i|p| n−1 ˆ hji hN −ji Pl − µ min , λ 2 2 l j−2 m=1 im l=1 2N 2N

2 i|p| Q

Rn−1 N 2(n−1) DP E DP E . aj (i1 , · · · , in−1 ) := Q n−1 l l i i − j k k l=1 k=1 k=1

p,α,β p,α,β ’s are defined by integrating the kernels κp,α,β consider (1.175). The Sj,ς To estimate Sj,ς j,ς hji hN −ji over Γj |p|-times. Since |Γj | = 2π min 2N 2 , 2N 2 , one gets n h i|p| X p,α,β hji hN −ji δ j, il Sj,ς (i) ≤ min N 2 , N 2

!

n X p,α,β il κj,ς (i) ≤ 2δ j,

l=1

! aj (i1 , · · · , in−1 ).

l=1

n ñ n Finally consider Kj,ς . From (1.173) one has Kj,ς (i, ι) ≤ 2n K j,ς (i) , and from (1.174) n X ñ il Kj,ς (i) ≤ δ j,

!

l=1

n

≤ C δ j,

X

aj (i1 , · · · , in−1 )

1

(α,β)∈Np1 ×Np2 |α|=r, |β|=s

r+s=n−(ς−1) p=(p1 ,p2 )∈N2 , |p|≤n

n X

X

cp1 cp2

! aj (i1 , · · · , in−1 ) ,

il

l=1

thus the first estimate of (1.179) follows. The proof of the second one is similar, and is omitted. n n Define now Kjn := Kj,1 + Kj,2 . Then

zjn (ˆb, a ˆ) =

Dj N n/2

X

Kjn (i, ι) ui1 ,ι1 . . . uin ,ιn ,

wjn (ˆb, a ˆ) =

(i,ι)∈∆n

Dj N n/2

X

Hjn (i, ι) ui1 ,ι1 . . . uin ,ιn ,

(i,ι)∈∆n

(1.180) where Hjn (i, ι) = Kjn (−i, ι). The second formula holds since for b, a real one has wn (b, a) = z n (b, a). Pn Corollary 1.75. Let ∆nj := {(i, ι) ∈ ∆n : l=1 il = j}. Then for 1 ≤ j ≤ bN/2c one has n n supp Kjn ⊆ ∆nj and supp KN ⊆ ∆ . Moreover −j −j

n

Kj n , ∆ j

n 2(n−1)

n

KN −j n ≤ R N , n−1 ∆−j hji

2 2 P where Kjn ∆n := supι1 ,··· ,ιn ∈{1,2} i1 +···+in =j Kjn (i, ι) . j

62

(1.181)

Proof. Just remark that

hji2 hki2 hk−ji2

≤4

1 hki2

+

1 hk−ji2

.

We prove now bounds on the map Z n (ˆb, a ˆ) := (z n (ˆb, a ˆ), wn (ˆb, a ˆ)). Lemma 1.76. There exists a constant C > 0, independent of N , such that for any s ≥ 0 and σ ≥ 0

n ˆ

n a|) ≤ C n N 2(n−1) k(b, a)kC s,σ , ∀ n ≥ 2. (1.182)

Z (|b|, |ˆ P s+1,σ

Proof. By formula (1.172) one has that for 1 ≤ j ≤ bN/2c X Dj n ˆ Kjn (i, ι) |ui1 ,ι1 | . . . |uin ,ιn |, a|) ≤ n/2 z j (|b|, |ˆ N n (i,ι)∈∆j

Dj n a|) ≤ n/2 z N −j (|ˆb|, |ˆ N

(1.183)

X

n KN −j (i, ι) |ui1 ,ι1 | . . . |uin ,ιn |.

(i,ι)∈∆n −j

Introduce Λ(i) := [i1 ] · · · [in ], where [ir ] = max(1, |ir |) ∀1 ≤ r ≤ n, and remark that for some constant R > 0 one has Rn sup Λ(i)−1 ≤ , ∀j ∈ Z. hji i1 +···+in =j Therefore, by Corollary 1.75, X 2

2 1 n ˆ a|) ≤ n Dj2 Kjn ∆n sup Λ(i)−2s [i1 ]2s |ui1 ,ι1 |2 . . . [in ]2s |uin ,ιn |2 , z j (|b|, |ˆ j N i1 +···+in =j (i,ι)∈∆n j X 2

1 n n

2 a|) ≤ n Dj2 KN sup Λ(i)−2s [i1 ]2s |ui1 ,ι1 |2 . . . [in ]2s |uin ,ιn |2 . z N −j (|ˆb|, |ˆ −j ∆n −j N i1 +···+in =−j n (i,ι)∈∆−j

Use now inequalities (1.181), the definition of Dj , the fact that e2σ|j| ≤ e2σ|i1 | · · · e2σ|in−1 | e2σ|j−i1 −···−in−1 | , and the bounds |ul,ιl | ≤ |ˆbl | + |ˆ al |, to deduce that, for any n ≥ 2, bN/2c 1 X 2(s+1) 2σ|j| [j] e ω N j=1

≤ N 4(n−1) ≤

j N

2 2 n n ˆ ˆ a|) + z N −j (|b|, |ˆ a|) z j (|b|, |ˆ

bN/2c C n X 2(2−n) 2σ|j| [j] e N n j=1

2n N 4(n−1) C n k(b, a)kC s,σ

X

[i1 ]2s |ui1 ,ι1 |2 . . . [in ]2s |uin ,ιn |2

(i,ι)∈∆n ±j

.

Since wn (ˆb, a ˆ) satisfies the same inequality, estimate (1.182) holds. Consider now the map (ˆb, a ˆ) 7→ dZ n (ˆb, a ˆ)∗ , where dZ n (ˆb, a ˆ)∗ is the adjoint of the differenn N −1 tial of Z . Explicitly, if ξ, η are vectors in C and h, g are vectors in CN such that (h, g) ≡ dZ n (ˆb, a ˆ)∗ (ξ, η), then the j th components of h and g are given by ! !! N −1 N −1 X X ∂zkn ˆ ∂wkn ˆ ∂zkn ˆ ∂wkn ˆ (hj , gj ) = (b, a ˆ)ξk + (b, a ˆ)ηk , (b, a ˆ)ξk + (b, a ˆ)ηk . (1.184) ∂ˆ aj ∂ˆ aj ∂ˆbj ∂ˆbj k=1

k=1

63

Denote by h, g the vectors of CN whose components are given by (hj , gj ) =

N −1 X

! ∂zkn ˆ ∂wkn ˆ (|b|, |ˆ a|)|ξk | + (|b|, |ˆ a|)|ηk | , ∂ˆbj ∂ˆbj

k=1

N −1 X k=1

!! ∂wkn ˆ ∂zkn ˆ (|b|, |ˆ a|)|ξk | + (|b|, |ˆ a|)|ηk | . ∂ˆ aj ∂ˆ aj (1.185)

We begin to study the case n = 2. Lemma 1.77. There exists a constant R > 0, independent of N , such that ∀s ≥ 0 , σ ≥ 0 one has

2 ˆ

(1.186) a|)∗ (|ξ|, |η|) s+2,σ ≤ RN 3 k(b, a)kC s,σ k(ξ, η)kP s,σ .

dZ (|b|, |ˆ C

Proof. By (1.159), one computes that the second order terms Z 2 = (z 2 , w2 ) are given by X ˆbl − 2 cos( (k−2l)π )ˆ ˆbk−l − 2 cos( kπ )ˆ zk2 (ˆb, a ˆ) = DNk a a /(λ02(k−2l) − λ02k ) l k−l N N l6=0

wk2 (ˆb, a ˆ)

=

Dk N

X 0 0 kπ ˆbN −l − 2 cos( (k−2l)π )ˆ ˆ a b − 2 cos( )ˆ a N −l l−k l−k /(λ2(k−2l) − λ2k ). N N l6=0

Let h, g be as in (1.185) with n = 2. Using the explicit expressions for zk2 and wk2 , one computes that for 0 ≤ j ≤ bN/2c N −1 |ˆ b | + 2|ˆ a | Dk (|ξk | + |ηk |) X k−j k−j 1 |hj | ≤ 0 N |λ2(k−2j) − λ02k | k=1 bN/2c |ˆ N −1 bk−j | + 2|ˆ ak−j | Dk (|ξk | + |ηk |) |ˆbk−j | + 2|ˆ ak−j | Dk (|ξk | + |ηk |) X X ≤N +N hk − jihji hN − k + jihji k=1 k=bN/2c+1 bN/2c |ˆ bk−j | + 2|ˆ ak−j | Dk (|ξk | + |ηk |) |ˆbN −k−j | + 2|ˆ aN −k−j | Dk (|ξN −k | + |ηN −k |) X ≤N + hk − jihji hk + jihji k=1 bN/2c ˆ ak−j | hki1/2 (|ξk | + |ηk |) |ˆbN −k−j | + 2|ˆ aN −k−j | hki1/2 (|ξN −k | + |ηN −k |) N 2 X |bk−j | + 2|ˆ ≤ + hji hk − jihki hk + jihki k=1

where in the last inequality we used that Dk ≤ N/hki1/2 . With analogous computations, one verifies that 1/2 ˆbk+j | + 2|ˆ ˆbj−k | + 2|ˆ | a | hki (|ξ | + |η |) | a | hki1/2 (|ξN −k | + |ηN −k |) 2 bN/2c X k+j k k j−k N + . |hN −j | ≤ hji hk + jihki hk − jihki k=1

Proceeding as in the proof of Lemma 1.70, one obtains that there exist constants C, C 0 > 0,

64

independent of N , such that bN/2c 1 X 2(s+2) 2σ|j| [j] e (|hj |2 + |hN −j |2 ) N j=0

≤ CN

3

N −1 X

!

2σ[k]N ˆ 2 (|bk | [k]2s Ne

2

+ |ˆ ak | )

N −1 X

!

2σ[l]N [l]N (|ξl |2 [l]2s Ne

2

+ |ηl | )

l=1

k=0 2

2

≤ C 0 N 6 k(b, a)kC s,σ k(ξ, η)kP s,σ

(1.187)

where in the last inequality we used that [l]N ≤ N ω Nl for l integer. One verifies that g satisfies the same inequality as (1.187). Thus estimate (1.186) follows from the following inequality:

N −1

2 1 X 2s+4 2σ[j]N

2 ˆ

a|)∗ (|ξ|, |η|) s+2,σ ≤ [j]N e |hj |2 + |gj |2 .

dZ (|b|, |ˆ N j=0 C

(1.188)

We study now dZ n (ˆb, a ˆ)∗ for n ≥ 3. Lemma 1.78. There exists a constant R > 0, independent of N , such that for every s ≥ 0, σ ≥ 0 and n ≥ 3

n ˆ

n−1 a|)∗ (|ξ|, |η|) s+2,σ ≤ Rn N 2n−1 k(b, a)kC s,σ k(ξ, η)kP s,σ . (1.189)

dZ (|b|, |ˆ C

Proof. Let h, g be as in (1.185). We concentrate on h only, the estimates for g being analogous. PN −1 ∂zn PN −1 ∂wn Write hj = k=1 ∂ˆbk ξk + k=1 ∂ˆb k ηk =: hj,1 + hj,2 . By (1.180) one gets that j

hj,1 =

1 N n/2

j

n X

An,l j (Dξ, u, . . . , u),

hj,2 =

l=1

1 N n/2

n X

Bjn,l (Dη, u, . . . , u)

l=1

where D is defined in (1.157), the multilinear map An,l j is defined by X An,l An,l j (h, u, . . . , u) = j (i, ι)ui1 ,ι1 . . . hil . . . uin ,ιn , (i,ι)∈∆n

Bjn,l is defined analogously but with kernel Bjn,l (i, ι), and finally An,l and Bjn,l are defined for j 1 ≤ j ≤ bN/2c by n An,l (i, ι) := K (i , . . . , i , j, i , . . . , i ), (ι , . . . , ι , 1, ι , . . . , ι ) , 1 l−1 l+1 n 1 l−1 l+1 n il j n An,l N −j (i, ι) := Kil (i1 , . . . , il−1 , −j, il+1 , . . . , in ), (ι1 , . . . , ιl−1 , 1, ιl+1 , . . . , ιn ) , n,l n,l while Bjn,l (i, ι) = An,l j (−i, ι) and BN −j (i, ι) = AN −j (−i, ι), see (1.180). By Corollary (1.75) it follows that n,l n supp An,l j = supp BN −j ≡ {(i, ι) : i1 + · · · + il−1 − il + il+1 + · · · + in = −j, ιl = 1} ⊆ ∆−j , n,l n supp An,l N −j = supp Bj ≡ {(i, ι) : i1 + · · · + il−1 − il + il+1 + · · · + in = j, ιl = 1} ⊆ ∆j .

65

Proceeding as in the proof of Corollary 1.75, one proves that there exists a constant R > 0, independent of N , such that (see [KP10])

max An,l j

∆n −j

1≤l≤n

, An,l N −j

n,l ,

B j n

∆n −j

∆j

∆j

n,l ,

B N −j n

≤

Rn N 2(n−1) , hji2

∀n ≥ 3 .

(1.190)

Thus h, defined in (1.185), satisfies |hj | ≤

1

n X

N n/2

n,l An,l j (|Dξ|, |u|, . . . , |u|) + Bj (|Dη|, |u|, . . . , |u|) ,

l=1

P n,l n,l where An,l j (h, u, . . . , u) = (i,ι)∈∆n Aj (i, ι) ui1 ,ι1 . . . hil . . . uin ,ιn , and Bj is defined in analogous way. Then, using (1.190) and arguing as in the proof of Lemma 1.76, one proves the estimate ! N −1 N −1 1 X 2(s+2) 2σ[j]N 1 X 2s 2σ[l]N 2 2(n−1) 2 n 4n−5 2 2 [j] e |hj | ≤ R N k(b, a)kC s,σ [l]N e Dl (|ξl | + |ηl | ) N j=0 N N l=1

n

≤R N

4n−2

2(n−1) k(b, a)kC s,σ

where in the last inequality we used that Dl2 ≤ inequality, thus estimate (1.189) follows.

N3 [l]2N

ω

2

k(ξ, η)kP s−1,σ ,

l N

. One verifies that g satisfies the same

We can finally prove property (Z4). Let s ≥ 0, σ ≥ 0 be fixed. By Lemma 1.76, 1.77 and 1.78, there exists C1 , C2 , ∗ > 0, independent of N , such that for every 0 < ≤ ∗ it holds that X

0

Z (b, a) s+1,σ ≤ sup kZ n (b, a)k s+1,σ sup k(b,a)kCs,σ ≤/N 2

P

≤

X

Rn N 2(n−1)

n≥2

sup k(b,a)kCs,σ ≤/N 2

P

2 n≥2 k(b,a)kCs,σ ≤/N

X

0

dZ (b, a)∗ s,σ s+2,σ ≤ L(P ,C )

sup

X

Rn N 2n−1

C 1 2 n ≤ , N 2n N2

2 n≥2 k(b,a)kCs,σ ≤/N

≤

n≥2

66

kdZ n (b, a)∗ kL(P s,σ , C s+2,σ )

n−1 ≤ C2 N . N 2(n−1)

Chapter 2

One smoothing properties of the KdV flow on R 1

Introduction

In the last decades the problem of a rigorous analysis of the theory of infinite dimensional integrable Hamiltonian systems in 1-space dimension has been widely studied. These systems come up in two setups: (i) on compact intervals (finite volume) and (ii) on infinite intervals (infinite volume). The dynamical behaviour of the systems in the two setups have many similar features, but also distinct ones, mostly due to the different manifestation of dispersion. The analysis of the finite volume case is now quite well understood. Indeed, Kappeler with collaborators introduced a series of methods in order to construct rigorously Birkhoff coordinates (a cartesian version of action-angle variables) for 1-dimensional integrable Hamiltonian PDE’s on T. The program succeeded in many cases, like Korteweg-de Vries (KdV) [KP03], defocusing and focusing Nonlinear Schrödinger (NLS) [GK14, KLTZ09]. In each case considered, it has been proved that there exists a real analytic symplectic diffeomorphism, the Birkhoff map, between two scales of Hilbert spaces which conjugate the nonlinear dynamics to a linear one. An important property of the Birkhoff map Φ of the KdV on T and its inverse Φ−1 is the semilinearity, i.e., the nonlinear part of Φ respectively Φ−1 is 1-smoothing. A local version of this result was first proved by Kuksin and Perelman [KP10] and later extended globally by Kappeler, Schaad and Topalov [KST13]. It plays an important role in the perturbation theory of KdV – see [Kuk10] for randomly perturbed KdV equations and [ET13b] for forced and weakly damped problems. The semi-linearity of Φ and Φ−1 can be used to prove 1-smoothing properties of the KdV flow in the periodic setup [KST13]. The analysis of the infinite volume case was developed mostly during the ’60-’70 of the last century, starting from the pioneering works of Gardner, Greene, Kruskal and Miura [GGKM67, GGKM74] on the KdV on the line. In these works the authors showed that the KdV can be integrated by a scattering transform which maps a function q, decaying sufficiently fast at infinity, into the spectral data of the operator L(q) := −∂x2 + q. Later, similar results were obtained by Zakharov and Shabat for the NLS on R [ZS71], by Ablowitz, Kaup, Newell and Segur for the Sine-Gordon equation [AKNS74], and by Flaschka for the Toda lattice with infinitely many particles [Fla74].

67

Furthermore, using the spectral data of the corresponding Lax operators, action-angle variables were (formally) constructed for each of the equations above [ZF71, ZM74, McL75b, McL75a]. See also [NMPZ84, FT87, AC91] for monographs about the subject. Despite so much work, the analytic properties of the scattering transform and of the action-angle variables in the infinite volume setup are not yet completely understood. In the present paper we discuss these properties, at least for a special class of potentials. The aim of this paper is to show that for the KdV on the line, the scattering map is an analytic perturbation of the Fourier transform by a 1-smoothing nonlinear operator. With the applications we have in mind, we choose a setup for the scattering map so that the spaces considered are left invariant under the KdV flow. Recall that the KdV equation on R ( ∂t u(t, x) = −∂x3 u(t, x) − 6u(t, x)∂x u(t, x) , (2.1) u(0, x) = q(x) , is globally in time well-posed in various function spaces such as the Sobolev spaces H N ≡ H N (R, R), N ∈ Z≥2 ( e.g. [BS75, Kat79, KPV93]), as well as on the weighted spaces H 2N ∩ L2M , with integers N ≥ M ≥ 1 [Kat66], endowed with the norm k · kH 2N + k · kL2M . Here L2M ≡ L2M (R, C) denotes the R 21 ∞ < ∞. space of complex valued L2 -functions satisfying kqkL2M := −∞ (1 + |x|2 )M |q(x)|2 dx Introduce for q ∈ L2M with M ≥ 4 the Schrödinger operator L(q) := −∂x2 + q with domain where, for any integer N ∈ Z≥0 , HCN := H N (R, C). For k ∈ R denote by f1 (q, x, k) and f2 (q, x, k) the Jost solutions, i.e. solutions of L(q)f = k 2 f with asymptotics f1 (q, x, k) ∼ eikx , x → ∞, f2 (q, x, k) ∼ e−ikx , x → −∞. As fi (q, ·, k), fi (q, ·, −k), i = 1, 2, are linearly independent for k ∈ R \ {0}, one can find coefficients S(q, k), W (q, k) such that for k ∈ R \ {0} one has

HC2 ,

W (q, k) S(q, −k) f1 (q, x, k) + f1 (q, x, −k) , 2ik 2ik S(q, k) W (q, k) f1 (q, x, k) = f2 (q, x, k) + f2 (q, x, −k) . 2ik 2ik f2 (q, x, k) =

(2.2)

It’s easy to verify that the functions W (q, ·) and S(q, ·) are given by the wronskian identities W (q, k) := [f2 , f1 ] (q, k) := f2 (q, x, k)∂x f1 (q, x, k) − ∂x f2 (q, x, k)f1 (q, x, k) ,

(2.3)

and S(q, k) := [f1 (q, x, k), f2 (q, x, −k)] ,

(2.4)

which are independent of x ∈ R. For q ∈ Q the functions S(q, k) and W (q, k) are related to the more often used reflection coefficients r± (q, k) and transmission coefficient t(q, k) by the formulas r+ (q, k) =

S(q, −k) , W (q, k)

r− (q, k) =

S(q, k) , W (q, k)

t(q, k) =

2ik W (q, k)

∀ k ∈ R \ {0} .

(2.5)

It is well known that for q real valued the spectrum of L(q) consists of an absolutely continuous part, given by [0, ∞), and a finite number of eigenvalues referred to as bound states, −λn < · · · < −λ1 < 0 (possibly none). Introduce the set Q := q : R → R , q ∈ L24 : W (q, 0) 6= 0, q without bound states . (2.6) 68

We remark that the property W (q, 0) 6= 0 is generic. In the sequel we refer to elements in Q as generic potentials without bound states. Finally we define QN,M := Q ∩ H N ∩ L2M ,

N ∈ Z≥0 ,

M ∈ Z≥4 .

We will see in Lemma 2.25 that for any integers N ≥ 0, M ≥ 4, QN,M is open in H N ∩ L2M . Our main theorem analyzes the properties of the scattering map q 7→ S(q, ·) which is known to linearize the KdV flow [GGKM74]. To formulate our result on the scattering map in more details let S denote the set of all functions σ : R → C satisfying (S1) σ(−k) = σ(k),

∀k ∈ R;

(S2) σ(0) > 0. For M ∈ Z≥1 define the real Banach space HζM := {f ∈ HCM −1 :

f (k) = f (−k),

ζ∂kM f ∈ L2 } ,

(2.7)

where ζ : R → R is an odd monotone C ∞ function with ζ(k) = k for |k| ≤ 1/2

and

ζ(k) = 1 for k ≥ 1 .

(2.8)

The norm on HζM is given by

2 2 2 kf kH M := kf kH M −1 + ζ∂kM f L2 . ζ

C

For any N, M ∈ Z≥0 let S M,N := S ∩ HζM ∩ L2N .

(2.9)

Different choices of ζ, with ζ satisfying (2.8), lead to the same Hilbert space with equivalent norms. We will see in Lemma 2.26 that for any integers N ≥ 0, M ≥ 4, S M,N is an open subset of HζM ∩L2N . R +∞ Moreover let F± be the Fourier transformations defined by F± (f ) = −∞ e∓2ikx f (x) dx. In this setup, the scattering map S has the following properties – see Appendix B for a discussion of the notion of real analytic. Theorem 2.1. For any integers N ≥ 0, M ≥ 4, the following holds: (i) The map S : QN,M → S M,N ,

q 7→ S(q, ·)

is a real analytic diffeomorphism. −1 (ii) The maps A := S − F− and B := S −1 − F− are 1-smoothing, i.e.

A : QN,M → HζM ∩ L2N +1

and

Furthermore they are real analytic maps.

69

B : S M,N → H N +1 ∩ L2M −1 .

As a first application of Theorem 2.1 we prove analytic properties of the action variable for the KdV on the line. For a potential q ∈ Q, the action-angle variable were formally defined for k 6= 0 by Zakharov and Faddeev [ZF71] as the densities k |S(q, k)|2 , θ(q, k) := arg (S(q, k)) , k ∈ R \ {0} . (2.10) I(q, k) := log 1 + π 4k 2 We can write the action as I(q, k) := −

k log π

4k 2 4k 2 + S(q, k)S(q, −k)

,

k ∈ R \ {0} .

(2.11)

By Theorem 2.1, S(q, ·) ∈ S , thus property (S2) implies that limk→0 I(q, k) exists and equals 0. Furthermore, by (S1), the action I(q, ·) is an odd function in k, and strictly positive for k > 0. Thus we will consider just the case k ∈ [0, +∞). The properties of I(q, ·) for k near 0 and k large are described separately. Corollary 2.2. For any integers N ≥ 0, M ≥ 4, the maps QN,M → L12N +1 ([1, +∞), R) ,

q 7→ I(q, ·)|[1,∞)

and QN,M → H M ([0, 1], R) ,

q 7→ I(q, ·)|[0,1] +

k ln π

4k 2 4(k 2 + 1)

are real analytic. Finally we compare solutions of (2.1) to solutions of the Cauchy problem for the Airy equation on R, ( ∂t v(t, x) = −∂x3 v(t, x) (2.12) v(0, x) = p(x) Being a linear equation with constant coefficients, one sees that the Airy equation is globally in time well-posed on H N and H 2N ∩ L2M , with integers N ≥ M ≥ 1 (see Remark 2.41 below). t t (p) := v(t, ·) respectively UKdV (q) := u(t, ·). Our Denote the flows of (2.12) and (2.1) by UAiry 2N 2 third result is to show that for q ∈ H ∩ LM with no bound states and W (q, 0) 6= 0, the difference t t UKdV (q) − UAiry (q) is 1-smoothing, i.e. it takes values in H 2N +1 . More precisely we prove the following theorem. Theorem 2.3. Let N , M be integers with N ≥ 2M ≥ 8. Then the following holds true: (i) QN,M is invariant under the KdV flow. t t (ii) For any q ∈ QN,M the difference UKdV (q) − UAiry (q) takes values in H N +1 ∩ L2M . Moreover the map

QN,M × R≥0 →H N +1 ∩ L2M ,

t t (q, t) 7→ UKdV (q) − UAiry (q)

is continuous and for any fixed t real analytic in q.

70

Outline of the proof: In Section 2 we study analytic properties of the Jost functions fj (q, x, k), j = 1, 2, in appropriate Banach spaces. We use these results in Section 3 to prove the direct scattering part of Theorem 2.1. The inverse scattering part of Theorem 2.1 is proved in Section 4. Finally in Section 5 we prove Corollary 2.2 and Theorem 2.3. Related works: As we mentioned above, this paper is motivated in part from the study of the 1-smoothing property of the KdV flow in the periodic setup, established recently in [BIT11, ET13a, KST13]. In [KST13] the one smoothing property of the Birkhoff map has been exploited to prove t t that for q ∈ H N (T, R), N ≥ 1, the difference UKdV (q) − UAiry (q) is bounded in H N +1 (T, R) with a bound which grows linearly in time. Kappeler and Trubowitz [KT86, KT88] studied analytic properties of the scattering map S between weighted Sobolev spaces. More precisely, define the spaces H n,α := f ∈ L2 : xβ ∂xj f ∈ L2 , 0 ≤ j ≤ n, 0 ≤ β ≤ α , H]n,α := f ∈ H n,α : xβ ∂xn+1 f ∈ L2 , 1 ≤ β ≤ α . In [KT86], Kappeler and Trubowitz showed that the map q 7→ S(q, ·) is a real analytic diffeomorphism from Q ∩ H N,N to S ∩ H]N −1,N , N ∈ Z≥3 . They extend their results to potentials with finitely many bound states in [KT88]. Unfortunately, Q ∩ H N,N is not left invariant under the KdV flow. Results concerning the 1-smoothing property of the inverse scattering map were obtained previously in [Nov96], where it is shown that for a potential q in the space W n,1 (R, R) of real-valued functions with weak derivatives up to order n in L1 Z 1 e−2ikx χc (k)2ikr+ (q, k)dk ∈ W n+1,1 (R, R) . q(x) − π R Here c is an arbitrary number with c > kqkL1 and χc (k) = 0 for |k| ≤ c , χc (k) = |k| − c for c ≤ |k| ≤ c + 1, and 1 otherwise. The main difference between the result in [Nov96] and ours concerns the function spaces considered. For the application to the KdV we need to choose function spaces such as H N ∩ L2M for which KdV is well posed. To the best of our knowledge it is not known if KdV is well posed in W n,1 (R, R). Furthermore in [Nov96] the question of analyticity of the map q 7→ r+ (q) and its inverse is not addressed. We remark that Theorem 2.1 treats just the case of regular potentials. In [FHMP09, HMP11] a special class of distributions is considered. In particular the authors study Miura potentials −1 q ∈ Hloc (R, R) such that q = u0 + u2 for some u ∈ L1 (R, R) ∩ L2 (R, R), and prove that the map q 7→ r+ is bijective and locally bi-Lipschitz continuous between appropriate spaces. Finally we point out the work of Zhou [Zho98], in which L2 -Sobolev space bijectivity for the scattering and inverse scattering transforms associated with the ZS-AKNS system are proved.

2

Jost solutions

In this section we assume that the potential q is complex-valued. Often we will assume that q ∈ L2M with M ∈ Z≥4 . Consider the normalized Jost functions m1 (q, x, k) := e−ikx f1 (q, x, k) and

71

m2 (q, x, k) := eikx f2 (q, x, k) which satisfy the following integral equations Z +∞ Dk (t − x) q(t) m1 (q, t, k)dt m1 (q, x, k) = 1 + Zxx m2 (q, x, k) = 1 + Dk (x − t) q(t) m2 (q, t, k)dt

(2.13) (2.14)

−∞

Ry where Dk (y) := 0 e2iks ds. The purpose of this section is to analyze the solutions of the integral equations (2.13) and (2.14) in spaces needed for our application to KdV. We adapt the corresponding results of [KT86] to these spaces. As (2.13) and (2.14) are analyzed in a similar way we concentrate on (2.13) only. For simplicity we write m(q, x, k) for m1 (q, x, k). For 1 ≤ p ≤ ∞, M ≥ 1 and a ∈ R, 1 ≤ α < ∞, 1 ≤ β ≤ ∞ we introduce the spaces n o β LpM := f : R → C : hxiM f ∈ Lp , Lα x≥a L := f : [a, +∞) × R → C : kf kLα Lβ < +∞ x≥a

where hxi := (1 + x2 )1/2 , Lp is the standard Lp space, and Z +∞ 1/α α kf kLα Lβ := kf (x, ·)kLβ dx x≥a

a

0 Lβ := whereas for α = ∞, kf kL∞ Lβ := supx≥a kf (x, ·)kLβ . We consider also the space Cx≥a x≥a β C 0 [a, +∞), Lβ with kf kC 0 Lβ := supx≥a kf (x, ·)kLβ < ∞. We will use also the space Lα x≤a L of x≥a R 1/α a α . Moreover functions f : (−∞, a] × R → C with finite norm kf kLα Lβ := kf (x, ·)kLβ dx −∞ x≤a

given any Banach spaces X and Y we denote by L(X, Y ) the Banach space of linear bounded operators from X to Y endowed with the operator norm. If X = Y , we simply write L(X). For the notion of an analytic map between complex Banach spaces we refer to Appendix B. We begin by stating a well known result about the properties of m. Theorem 2.4 ( [DT79]). Let q ∈ L11 . For each k, Im k ≥ 0, the integral equation +∞ Z m(x, k) = 1 + Dk (t − x)q(t)m(t, k)dt ,

x∈R

x

has a unique solution m ∈ C 2 (R, C) which solves the equation m00 +2ikm0 = q(x)m with m(x, k) → 1 as x → +∞. If in addition q is real valued the function m satisfies the reality condition m(q, k) = m(q, −k). Moreover, there exists a constant K > 0 which can be chosen uniformly on bounded subsets of L11 such that the following estimates hold for any x ∈ R (i) |m(x, k) − 1| ≤ eη(x)/|k| η(x)/|k|,

k 6= 0;

+∞ R (ii) |m(x, k) − 1| ≤ K (1 + max(−x, 0)) (1 + |t|)|q(t)|dt /(1 + |k|); x

(iii) |m0 (x, k)| ≤ K1

+∞ R (1 + |t|)|q(t)|dt /(1 + |k|) x

72

where η(x) =

+∞ R

|q(t)|dt. For each x, m(x, k) is analytic in Im k > 0 and continuous in Im k ≥ 0.

x

In particular, for every x fixed, k 7→ m(x, k) − 1 ∈ H 2+ , where H 2+ is the Hardy space of functions +∞ R analytic in the upper half plane such that supy>0 |h(k + iy)|2 dk < ∞. −∞

Estimates on the Jost functions. Proposition 2.5. For any q ∈ L2M with M ≥ 2, a ∈ R and 2 ≤ β ≤ +∞, the solution m(q) of 0 0 (2.13) satisfies m(q) − 1 ∈ Cx≥a Lβ ∩ L2x≥a L2 . The map L2M 3 q 7→ m(q) − 1 ∈ Cx≥a Lβ ∩ L2x≥a L2 is analytic. Moreover there exist constants C1 , C2 > 0, only dependent on a, β, such that kqk 1 kqk 1 km(q) − 1kC 0 Lβ ≤ C1 e L1 kqkL2 , km(q) − 1kL2 L2 ≤ C2 kqkL2 1 + kqkL2 e L1 . 1

x≥a

2

x≥a

3/2

(2.15) Remark 2.6. In comparison with [KT86], the novelty of Proposition 2.5 consists in the choice of spaces. To prove Proposition 2.5 we first need to establish some auxiliary results. Lemma 2.7.

(i) For any q ∈ L11 , a ∈ R and 1 ≤ β ≤ +∞, the linear operator

K(q) :

0 Cx≥a Lβ

→

0 Cx≥a Lβ ,

+∞ Z f→ 7 K(q)[f ](x, k) := Dk (t − x)q(t)f (t, k)dt

(2.16)

x

is bounded. Moreover for any n ≥ 1, the nth composition K(q)n satisfies kK(q)n kL(C 0 C

n

x≥a

n kqkL1 1

Lβ )

≤

/n! where C > 0 is a constant depending only on a. 0 (ii) The map K : L11 → L Cx≥a Lβ , q 7→ K(q), is linear and bounded, and Id − K is invertible. More precisely, −1 −1 0 (Id − K) : L11 → L Cx≥a Lβ , q 7→ (Id − K(q))

CkqkL1

−1 1. is analytic and (Id − K) 1 0 ≤e L(L1 ,Cx≥a Lβ ) Proof. Let h ∈ Lα with α1 + β1 = 1. Using |Dk (t − x)| ≤ |t − x|, one has +∞ +∞ Z Z h(k)K(q)[f ](x, k)dk ≤ dt |t − x||q(t)| kf (t, ·)kLβ khkLα −∞ x   +∞ Z |t − a||q(t)|dt kf kC 0 Lβ khkLα , ≤ x≥a

a

and hence kK(q)kL(C 0

Lβ ) x≥a

≤

+∞ R a

|t − a||q(t)|dt ≤ C kqkL1 , where C > 0 is a constant depending 1

just on a. To compute the norm of the iteration of the map K(q) it’s enough to proceed as above and 73

n

exploit the fact that the integration in t is over a simplex, yielding kK(q)n kC 0 Lβ ≤ C n kqkL1 /n! 1 x≥a −1 P n for any n ≥ 1. Therefore the Neumann series of the operator Id − K(q) = n≥0 K(q) 0 converges absolutely in L Cx≥a Lβ . Since K(q) is linear and bounded in q, the analyticity and, by item (i), the claimed estimate for (Id − K)−1 follow. Lemma 2.8. Let a ∈ R. (i) For any q ∈ L23/2 , K(q) defines a bounded linear operator L2x≥a L2 → L2x≥a L2 . Moreover the nth composition K(q)n satisfies kK(q)n kL(L2

n−1

x≥a

L2 )

≤ C n kqkL2 kqkL1 /(n − 1)! 3/2

1

where C > 0 depends only on a. (ii) The map K : L23/2 → L L2x≥a L2 , −1

(Id − K)

q 7→ K(q) is linear and bounded; the map

: L23/2 → L L2x≥a L2

−1 is analytic and (Id − K)

L(L23/2 ,L2x≥a L2 )

−1

q 7→ (Id − K(q))

kqk 1 ≤ C 1 + kqkL2 e L1 . 3/2

Proof. Proceeding as in the proof of the previous lemma, one gets for x ≥ a the estimate kK(q)[f ](x, ·)kL2

+∞ +∞ Z 1/2 Z (t − x)2 |q(t)|2 dt kf kL2 ≤ |t − x||q(t)| kf (t, ·)kL2 dt ≤

x≥a

L2

,

x

x

from which it follows that 2

kK(q)[f ]kL2

x≥a

L2

+∞

1/2

Z

2 2

≤ (t − x) |q(t)| dt kf kL2 L2 ≤ C kqkL2 kf kL2 L2

x≥a x≥a 3/2

1 x

Lx≥a

proving item (i). To estimate the composition K(q)n viewed as an operator on L2x≥a L2 , remark that Z n kK(q) [f ](x, ·)kL2 ≤ |t1 − x||q(t1 )| · · · |tn − tn−1 ||q(tn )| kf (tn , ·)kL2 dt x≤t1 ≤...≤tn

Z ≤

|t1 − x||q(t1 )| · · · |tn−1 − tn−2 ||q(tn−1 )|

dtn (tn − tn−1 )2 |q(tn )|2

tn−1

x≤t1 ≤...≤tn +∞ Z 1/2 ≤ (t − x)2 |q(t)|2 dt kf kL2

x≥a

x

+∞ Z

+∞ Z n−1 |t − x||q(t)| dt /(n − 1)! . 2 L x

74

1/2

kf kL2

x≥a

L2

dt

Therefore kK(q)n [f ]kL2

x≥a

1/2

+∞

Z n−1 C n−1 kqkL1

2 2 1

kf k (t − x) |q(t)| dt ≤ L2x≥a L2

(n − 1)!

1

L2

x

Lx≥a

n−1

≤ kqkL2 kf kL2

x≥a

3/2

L2

C

n

kqkL1 1

(n − 1)!

from which item (i) follows. Item (ii) is then proved as in the previous Lemma. Note that for f ≡ 1, the expression in (2.16) of K(q)[f ], K(q)[1](x, k) =

+∞ R

Dk (t − x) q(t) dt is

x

well defined. 0 Lemma 2.9. For any 2 ≤ β ≤ +∞ and a ∈ R, the map L22 3 q 7→ K(q)[1] ∈ Cx≥a Lβ ∩ L2x≥a L2 is analytic. Furthermore

kK(q)[1]kC 0

x≥a

Lβ

≤ C1 kqkL2 ,

kK(q)[1]kL2

2

x≥a

L2

≤ C2 kqkL2 , 2

where C1 , C2 > 0 are constants depending on a and β. Proof. Since the map q 7→ K(q)[1] is linear in q, it suffices to prove its continuity in q. Moreover, it is enough to prove the result for β = 2 and β = +∞ as the general case then follows by interpolation. For any k ∈ R, the bound |Dk (y)| ≤ |y| shows that the map k 7→ Dk (y) is in L∞ . Thus kK(q)[1](x, ·)kL∞

+∞ +∞ Z Z ≤ (t − x)|q(t)|dt ≤ |t − a||q(t)| dt ≤ C kqkL1 , 1

x

a

where C > 0 is a constant depending only on a ∈ R. The claimed estimate follows by noting that kqkL1 ≤ C kqkL2 . 1 2 1 Using that for |k| ≥ 1, |Dk (y)| ≤ |k| , one sees that k 7→ Dk (y) is L2 -integrable. Hence k 7→ Dk (t − x)D−k (s − x) is integrable. Actually, since the Fourier transform F+ (Dk (y)) in the kvariable of the function k 7→ Dk (y) is the function η 7→ 1[0,y] (η), by Plancherel’s Theorem Z

∞

Dk (t − x)Dk (s − x) dk = −∞

1 π

Z

∞

−∞

1[0,t−x] (η)1[0,s−x] (η) dη =

1 min(t − x, s − x). π

For any x ≥ a one thus has 2 kK(q)[1](x, ·)kL2

Z

∞

K(q)[1](x, ·) · K(q)[1](x, ·) dk

= −∞

ZZ =

+∞ Z dt ds q(t) q(s) Dk (t − x)D−k (s − x) dk . −∞

[x,∞)×[x,∞)

75

and hence 2 kK(q)[1](x, ·)kL2

2 ≤ π

+∞ +∞ +∞ Z Z s Z Z 2 2 |q(s)|ds ≤ |t − a| |q(t)| dt ≤ C kqkL2 , (t − x)|q(t)| ds |q(s)| 1 π a t

x

a

(2.17) where the last inequality follows from the Hardy-Littlewood inequality. The continuity in x follows from Lebesgue convergence Theorem. To prove the second inequality, start from the second term in (2.17) and change the order of integration to obtain

+∞ +∞ +∞

Z Z Z Zs

2

≤ |q(s)| (s−a)2 |q(s)|ds ≤ C kqkL2 kqkL2 . |q(s)|ds |t − a||q(t)| kK(q)[1]kL2 L2 ≤ 1 2 x≥a

1

x

t

Lx≥a

a

a

Proof of Proposition 2.5. Formally, the solution of equation (2.13) is given by −1 K(q)[1]. m(q) − 1 = Id − K(q)

(2.18)

0 By Lemma 2.7, 2.8, 2.9 it follows that the r.h.s. of (2.18) is an element of Cx≥a Lβ ∩ L2x≥a L2 , 2 ≤ β ≤ ∞, and analytic as a function of q, since it is the composition of analytic maps.

Properties of ∂kn m(q, x, k) for 1 ≤ n ≤ M − 1. In order to study ∂kn m(q, x, k), we deduce from (2.13) an integral equation for ∂kn m(q, x, ·) and solve it. Recall that for any M ∈ Z≥0 , HCM ≡ H M (R, C) denotes the Sobolev space of functions {f ∈ L2 | fˆ ∈ L2M }. The result is summarized in the following Proposition 2.10. Fix M ∈ Z≥4 and a ∈ R. For any integer 1 ≤ n ≤ M − 1 the following holds: (i) for q ∈ L2M and x ≥ a fixed, the function k 7→ m(q, x, k) − 1 is in HCM −1 ; 0 L2 is analytic. Moreover k∂kn m(q)kC 0 (ii) the map L2M 3 q 7→ ∂kn m(q) ∈ Cx≥a

x≥a

where K can be chosen uniformly on bounded subsets of

L2

≤ K kqkL2 , M

L2M .

Remark 2.11. In [CK87b] it is proved that if q ∈ L1M −1 then for every x ≥ a fixed the map k 7→ m(q, x, k) is in C M −2 ; note that since L2M ⊂ L1M −1 , we obtain the same regularity result by Sobolev embedding theorem. To prove Proposition 2.10 we first need to derive some auxiliary results. Assuming that m(q, x, ·)−1 has appropriate regularity and decay properties, the nth derivative ∂kn m(q, x, k) satisfies the following integral equation ∂kn m(q, x, k)

+∞ n Z X n = ∂kj Dk (t − x) q(t) ∂kn−j m(q, t, k) dt . j j=0 x

76

(2.19)

To write (2.19) in a more convenient form introduce for 1 ≤ j ≤ n and q ∈ L2n+1 the operators Kj (q) :

0 Cx≥a L2

→

0 Cx≥a L2 ,

+∞ Z f→ 7 Kj (q)[f ](x, k) := ∂kj Dk (t − x) q(t) f (t, k) dt

(2.20)

x

leading to

  n−1 X n Id − K(q) ∂kn m(q) =  Kj (q)[∂kn−j m(q)] + Kn (q)[m(q) − 1] + Kn (q)[1] . j j=1

(2.21)

In order to prove the claimed properties for ∂kn m(q) we must show in particular that the r.h.s. of 0 (2.21) is in Cx≥a L2 . This is accomplished by the following Lemma 2.12. Fix M ∈ Z≥4 and a ∈ R. Then there exists a constant C > 0, depending only on a, M , such that the following holds: (i) for any integers 1 ≤ n ≤ M − 1 0 L2 is analytic, and kKn (q)[1]kC 0 L2 ≤ C kqkL2 . (i1) the map L2M 3 q 7→ Kn (q)[1] ∈ Cx≥a M x≥a 2 2 0 2 2 (i2) the map LM 3 q 7→ Kn (q) ∈ L Lx≥a L , Cx≥a L is analytic. Moreover

kKn (q)[f ]kC 0

x≥a

L2

≤ kqkL2 kf kL2 M

x≥a

L2

.

0 (ii) For any 1 ≤ n ≤ M − 2, the map L2M 3 q 7→ Kn (q) ∈ L Cx≥a L2 is analytic. Moreover one has kKn (q)[f ]kC 0 L2 ≤ C kqkL2 kf kC 0 L2 . M

x≥a

x≥a

(iii) As an application of item (i) and (ii), for any integers 1 ≤ n ≤ M − 1 the map L2M 3 q 7→ 0 Kn (q)[m(q) − 1] ∈ Cx≥a L2 is analytic, and kKn (q)[m(q) − 1]kC 0

x≥a

2

L2

≤ K00 kqkL2 , M

where K00 > 0 can be chosen uniformly on bounded subsets of L2M . Proof. First, remark that all the operators q 7→ Kn (q) are linear in q, therefore the continuity in q implies the analyticity in q. We begin proving item (i). +∞ R

∂kn Dk (t − x) q(t) dt and compute the Fourier transform F+ (ϕ(x, ·)) with R∞ respect to the k variable for x ≥ a fixed, which we denote by ϕ(x, ˆ ξ) ≡ −∞ dk eikξ ϕ(x, k). Explicitly

(i1) Let ϕ(x, k) :=

x

+∞ +∞ +∞ Z Z Z 2ikξ n ϕ(x, ˆ ξ) = dt q(t) dk e ∂k Dk (t − x) = q(t) ξ n 1[0,t−x] (ξ) dt. x

−∞

x

77

By Parseval’s Theorem kϕ(x, ·)kL2 = one has 2 kϕ(x, ˆ ·)kL2

√1 π

+∞ Z ϕ(x, ˆ ξ) ϕ(x, ˆ ξ) dξ = = −∞

kϕ(x, ˆ ·)kL2 . By changing the order of integration +∞ Z dt ds q(t) q(s) |ξ|2n 1[0,t−x] (ξ) 1[0,s−x] (ξ)dξ ≤

ZZ

−∞

[x,∞)×[x,∞)

+∞

Z

(t − a)n |q(s)|ds

t≥a

+∞ +∞ Z Z

2n+1 |q(s)| ds ≤ (t − a)n+1 q L2 dt |q(t)| |t − x| ≤2

t

t

x 2

≤ C kqkL2

n+1

L2t≥a

,

+∞

R n

|q(s)| ds where we used that by (A3) in Appendix A, (t − a)

t

≤ C kqkL2

n+1

L2t≥a

.

(i2) Let f ∈ L2x≥a L2 , and using |∂kn Dk (t − x)| ≤ 2n |t − x|n+1 it follows that kKn (q)[f ](x, ·)kL2

+∞ Z ≤C |q(t)| |t − x|n+1 kf (t, ·)kL2 dt ≤ C kqkL2

kf kL2

n+1

x≥a

L2

;

x

0 by taking the supremum in the x variable one has Kn (q) ∈ L L2x≥a L2 , Cx≥a L2 , where the continuity in x follows by Lebesgue’s convergence theorem. The map q 7→ Kn (q) is linear and continuous, therefore also analytic. 0 We prove now item (ii). Let g ∈ Cx≥a L2 . From kKn (q)[g](x, ·)kL2 ≤

+∞ R x

|q(t)| |t−x|n+1 kg(t, ·)kL2 dt

it follows that sup kKn (q)[g](x, ·)kL2 ≤ kgkC 0

x≥a

x≥a

L2

+∞ Z |q(t)| |t − a|n+1 dt ≤ C kgkC 0

x≥a

L2

kqkL2

n+2

,

a

which implies the claimed estimate. The analyticity follows from the linearity and continuity of the map q 7→ Kn (q). Finally we prove item (iii). By Proposition 2.5, the map L2n+1 3 q 7→ m(q) − 1 ∈ L2x≥a L2 is 0 analytic. By item (i2) above the bilinear map L2n+1 × L2x≥a L2 3 (q, f ) 7→ Kn (q)[f ] ∈ Cx≥a L2 is 2 analytic; since the composition of analytic maps is analytic, the map Ln+1 3 q 7→ Kn (q)[m(q)−1] ∈ 0 Cx≥a L2 is analytic. By (i2) and Proposition 2.5 one has kKn (q)[m(q) − 1]kC 0

x≥a

L2

≤ C kqkL2

n+1

km(q) − 1kL2

x≥a

2

L2

≤ K00 kqkL2

n+1

where K00 can be chosen uniformly on bounded subsets of L2M . Proof of Proposition (2.10). The proof is carried out by a recursive argument in n. We assume 0 that q 7→ ∂kr m(q) is analytic as a map from L2M to Cx≥a L2 for 0 ≤ r ≤ n − 1, and prove that 2 0 2 n LM → Cx≥a L : q 7→ ∂k m(q) is analytic, provided that n ≤ M − 1. The case n = 0 is proved in 78

Proposition 2.5. We begin by showing that for every x ≥ a fixed k 7→ ∂kn−1 m(q, x, k) is a function in H 1 , therefore it has one more (weak) derivative in the k-variable. We use the following characterization of H 1 function [Bre11]: f ∈ H 1 iff there exists a constant C > 0 such that kτh f − f kL2 ≤ C|h|,

∀h ∈ R,

(2.22)

where (τh f )(k) := f (k + h) is the translation operator. Moreover the constant C above can be chosen to be C = k∂k ukL2 . Starting from (2.21) (with n − 1 instead of n), an easy computation shows that for every x ≥ a fixed (τh )∂kn−1 m(q) ≡ ∂kn−1 m(q, x, k + h) satisfies the integral equation (Id − K(q))(τh ∂kn−1 m(q) − ∂kn−1 m(q)) Z +∞ (τh ∂kn−1 Dk (t − x) − ∂kn−1 Dk (t − x))q(t)(m(q, t, k + h) − 1) dt = x Z +∞ (τh ∂kn−1 Dk (t − x) − ∂kn−1 Dk (t − x))q(t) dt + x Z +∞ + (∂kn−1 Dk (t − x)) q(t) (m(q, t, k + h) − m(q, t, k)) dt +

x n−2 X j=1

Z

+∞

+ x +∞

Z +

x

Z n − 1 +∞ (τh ∂kj Dk (t − x) − ∂kj Dk (t − x))q(t)∂kn−1−j m(q, t, k + h) dt j x ∂kj Dk (t − x) q(t) (τh ∂kn−1−j m(q, t, k) − ∂kn−1−j m(q, t, k)) dt (τh Dk (t − x) − Dk (t − x)) q(t) ∂kn−1 m(q, t, k + h) dt.

(2.23) In order to estimate the term in the fourth line on the right hand side of the latter identity, use item (i1) of Lemma 2.12 and the characterization (2.22) of H 1 . To estimate all the remaining lines, use the induction hypothesis, the estimates of Lemma 2.12, the fact that the operator norm of (Id − K(q))−1 is bounded uniformly in k and the estimate j j τh ∂k Dk (t − x) − ∂k Dk (t − x) ≤ C|t − x|j+2 |h|, ∀h ∈ R, to deduce that for every n ≤ M − 1

τh ∂ n−1 m(q) − ∂ n−1 m(q) 2 ≤ C|h|, k k L

∀h ∈ R,

which is exactly condition (2.22). This shows that k 7→ ∂kn−1 m(q, x, k) admits a weak derivative in 0 L2 . Formula (2.19) is therefore justified. We prove now that the map L2M 3 q 7→ ∂kn m(q) ∈ Cx≥a L2 is analytic for 1 ≤ n ≤ M − 1. Indeed equation (2.21) and Lemma 2.12 imply that k∂kn m(q)kC 0 L2 x≥a

≤K

0

kqkL2 + M

2 kqkL2 M

+

n−1 X j=1

kqkL2 ∂kn−j m(q) M

0 Cx≥a L2

0 where K 0 can be chosen uniformly on bounded subsets of q in L2M . Therefore ∂kn m(q) ∈ Cx≥a L2 and n one gets recursively k∂k m(q)kC 0 L2 ≤ K kqkL2 , where K can be chosen uniformly on bounded x≥a

M

79

subsets of q in L2M . The analyticity of the map q 7→ ∂kn m(q) follows by formula (2.21) and the fact that composition of analytic maps is analytic. Properties of k∂kn m(q, x, k) for 1 ≤ n ≤ M . The analysis of the M th k-derivative of m(q, x, k) requires a separate attention. It turns out that the distributional derivative ∂kM m(q, x, ·) is not necessarily L2 -integrable near k = 0 but the product k∂kM m(q, x, ·) is. This is due to the fact that ∂kM Dk (x)q(x) ∼ xM +1 q(x) which might not be L2 -integrable. However, by integration by parts, it’s easy to see that k∂kM Dk (x)q(x) ∼ xM q(x) ∈ L2 . The main result of this section is the following Proposition 2.13. Fix M ∈ Z≥4 and a ∈ R. Then for every integer 1 ≤ n ≤ M the following holds: (i) for every q ∈ L2M and x ≥ a fixed, the function k 7→ k∂kn m(q, x, k) is in L2 ; 0 (ii) the map L2M 3 q 7→ k∂kn m(q) ∈ Cx≥a L2 is analytic. Moreover kk∂kn mkC 0

x≥a

L2

≤ K1 kqkL2

M

where K1 can be chosen uniformly on bounded subsets of L2M . Formally, multiplying equation (2.19) by k, the function k∂kn m(q) solves   n−1 X n ˜ j (q)[∂ n−j m(q)] + K ˜ n (q)[m(q) − 1] + K ˜ n (q)[1] K (Id − K(q)) (k∂kn m(q)) =  k j j=1

(2.24)

where we have introduced for 0 ≤ j ≤ M and q ∈ L2M the operators 0 0 ˜ j (q) : Cx≥a K L2 → Cx≥a L2 ,

˜ j (q)[f ](x, k) := f 7→ K

+∞ Z k∂kj Dk (t − x) q(t) f (t, k) dt.

(2.25)

x

We begin by proving that each term of the r.h.s. of (2.25) is well defined and analytic as a function of q. The following lemma is analogous to Lemma 2.12: Lemma 2.14. Fix M ∈ Z≥4 and a ∈ R. There exists a constant C > 0 such that the following holds: (i) for any integers 1 ≤ n ≤ M

˜ n (q)[1] ∈ C 0 L2 is analytic, and ˜ n (q)[1] K (i1) the map L2M 3 q 7→ K

x≥a

0 Cx≥a L2

(i2) the map L2M

≤ C kqkL2 ;

˜ n (q) ∈ L L2 L2 , C 0 L2 is analytic. Moreover 3q→ 7 K x≥a x≥a

˜

≤ C kqkL2 kf kL2 L2 ;

Kn (q)[f ] 0 2 M

Cx≥a L

x≥a

˜ j (q) ∈ L C 0 L2 is analytic, and (ii) for any 1 ≤ j ≤ M − 1 the map L2M 3 q 7→ K x≥a

˜

≤ C kqkL2 kf kC 0 L2 .

Kj (q)[f ] 0 Cx≥a L2

80

M

x≥a

M

(iii) As an application of item (i) and (ii) we get ˜ n (q)[m(q) − 1] ∈ C 0 L2 is analytic with (iii1) for any 1 ≤ n ≤ M , the map L2M 3 q 7→ K x≥a

˜

2 ≤ K10 kqkL2 , (2.26)

Kn (q)[m(q) − 1] 0 Cx≥a L2

M

where K10 can be chosen uniformly on bounded subsets of L2M ; ˜ j (q)[∂ n−j m(q)] ∈ C 0 L2 is analytic with (iii2) for any 1 ≤ j ≤ n − 1, the map L2M 3 q 7→ K x≥a k

˜ 2 n−j ≤ K20 kqkL2 , (2.27)

Kj (q)[∂k m(q)] 0 2 Cx≥a L

M

where K20 can be chosen uniformly on bounded subsets of L2M . ˜ n (q), 0 ≤ n ≤ M , are linear, it is enough to prove that these Proof. (i) Since the maps q 7→ K maps are continuous. (i1) Introduce ϕ(x, k) :=

+∞ R

k∂kn Dk (t − x) q(t) dt. The Fourier transform F+ (ϕ(x, ·) of ϕ

x

with respect to the k-variable is given by F+ (ϕ(x, ·)) ≡ ϕ(x, ˆ ξ), where +∞ +∞ +∞ Z Z Z −2ikξ n n−1 ϕ(x, ˆ ξ) = dt q(t) dk e k∂k Dk (t − x) = −(2i) dt q(t) ∂ξ (ξ n 1[0,t−x] (ξ)), x

−∞

x

where ∂ξ (ξ n 1[0,t−x] (ξ)) is to be understood in the distributional sense. By Parseval’s ˆ ·)kL2 . Let C0∞ be the space of smooth, compactly Theorem kϕ(x, ·)kL2 = √1π kϕ(x, supported functions. Since ∞ Z kϕ(x, ˆ ·)kL2 = sup χ(ξ) ϕ(x, ˆ ξ) dξ , ξ ∞ χ∈C0 kχkL2 ≤1 −∞

one computes ∞ +∞ +∞ t−x Z Z Z Z∞ Z n n χ(ξ) ϕ(x, ˆ ξ) dξ = dt q(t) χ(ξ) ∂ξ ξ 1[0,t−x] (ξ) dξ = dt q(t) dξ ξ ∂ξ χ(ξ) −∞ x −∞ x 0 +∞ +∞ t−x Z Z Z n ≤ dt q(t)χ(t − x)(t − x) + n dt q(t) dξ χ(ξ)ξ n−1 x x 0 +∞ t−x Z Z ≤ kqkL2 kχkL2 + n dt |q(t)||t − x|n−1 dξ |χ(ξ)| M x 0 t−x R Z dξ |χ(ξ)| +∞ ≤ kqkL2 kχkL2 + n dt |q(t)||t − x|n 0 ≤ C kqkL2M kχkL2 M |t − x| x 81

where the last inequality follows from Cauchy-Schwartz and Hardy inequality, and C > 0 is a constant depending on a and M . (i2) As |k∂kn Dk (t − x)| ≤ 2n |t − x|n by integration by parts, it follows that for some constant C > 0 depending only on a and M ,

˜

Kn (q)[f ](x, ·)

L2

+∞ Z ≤C |t − x|n |q(t)| kf (t, ·)kL2 dt ≤ C kqkL2 kf kL2 M

x≥a

L2

.

x

Now take the supremum over x ≥ a in the expression above and use Lebesgue’s dominated convergence theorem to prove item (i2). (ii) The claim follows by the estimate

˜

Kj (q)[f ](x, ·)

L2

+∞ Z ≤C |t − x|j |q(t)| kf (t, ·)kL2 dt ≤ C kqkL1 kf kC 0 j

x≥a

L2

x

and the remark that kqkL1 ≤ C kqkL2 for 0 ≤ j ≤ M − 1. j

M

0 (iii) By Propositions 2.5 and 2.10 the maps L2M 3 q 7→ m(q)−1 ∈ Cx≥a L2 ∩L2x≥a L2 and L2M 3 q 7→ 0 ∂kn−j m(q) ∈ Cx≥a L2 are analytic; by item (ii) for any 1 ≤ n ≤ M −1, the bilinear map (q, f ) 7→ ˜ n (q)[f ] is analytic from L2 ×C 0 L2 to C 0 L2 . Since the composition of two analytic maps K M x≥a x≥a ˜ n (q)[m(q) − 1], K ˜ j (q)[∂ n−j m(q)] ∈ C 0 L2 is again analytic, item (iii) follows. Moreover K x≥a k n since m(q, x, k) and ∂k m(q, x, k) are continuous in the x-variable. The estimate (2.26) follows from item (ii) and Proposition 2.5, 2.10.

Proof of Proposition 2.13. One proceeds in the same way as in the proof of Proposition 2.10. 0 Given any 1 ≤ n ≤ M , we assume that q 7→ k∂kr m(q) is analytic as a map from L2M to Cx≥a L2 for 0 2 2 n 1 ≤ r ≤ n − 1, and deduce that q 7→ k∂k m(q) is analytic as a map from LM to Cx≥a L and satisfies equation (2.24) (with r instead of n). We begin by showing that for every x ≥ a fixed, k 7→ k∂kn−1 m(q, x, k) is a function in H 1 . Our argument uses again the characterization (2.22) of H 1 . Arguing as for the derivation of (2.23) one

82

gets the integral equation (Id − K(q))(τh (k∂kn−1 m(q)) − k∂kn−1 m(q)) = +∞ Z = τh (k∂kn−1 Dk (t − x)) − k∂kn−1 Dk (t − x) q(t)(m(q, t, k + h) − 1) dt x +∞ Z

+

τh (k∂kn−1 Dk (t − x)) − k∂kn−1 Dk (t − x) q(t) dt

x +∞ Z + (k∂kn−1 Dk (t − x))q(t) (m(q, t, k + h) − m(q, t, k)) dt x

+

n−2 X j=1

+∞ Z n−1 τh (k∂kj Dk (t − x)) − k∂kj Dk (t − x) q(t) ∂kn−1−j m(q, t, k + h) dt j x

+∞ Z + k∂kj Dk (t − x) q(t) τh ∂kn−1−j m(q, t, k) − ∂kn−1−j m(q, t, k) dt x +∞ Z + (τh Dk (t − x) − Dk (t − x)) q(t) (k + h)∂kn−1 m(q, t, k + h) dt . x

Using the estimates |τh Dk (t − x) − Dk (t − x)| ≤ C|t − x|2 |h| and

j j τh (k∂k Dk (t − x)) − k∂k Dk (t − x) ≤ C|t − x|j+1 |h|,

∀h ∈ R ,

obtained by integration by parts, the characterization (2.22) of H 1 , the inductive hypothesis, estimates of Lemma 2.12 and Lemma 2.8 one deduces that for every n ≤ M

τh (k∂ n−1 m(q)) − k∂ n−1 m(q) 2 ≤ C|h|, ∀h ∈ R. k k L This shows that k 7→ k∂kn−1 m(q, x, k) admits a weak derivative in L2 . Since k∂kn m(q, x, k) = ∂k (k∂kn−1 m(q, x, k)) − ∂kn−1 m(q, x, k) , the estimate above and Proposition 2.10 show that k 7→ k∂kn m(q, x, k) is an L2 function. Formula (2.19) is therefore justified. The proof of the analyticity of the map q 7→ k∂kn m(q) is analogous to the one of Proposition 2.10 and it is omitted. Analysis of ∂x m(q, x, k). Introduce a odd smooth monotone function ζ : R → R with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1. We prove the following Proposition 2.15. Fix M ∈ Z≥4 and a ∈ R. Then the following holds:

83

0 (i) for any integer 0 ≤ n ≤ M − 1, the map L2M 3 q 7→ ∂kn ∂x m(q) ∈ Cx≥a L2 is analytic, and n k∂k ∂x m(q)kC 0 L2 ≤ K2 kqkL2 where K2 can be chosen uniformly on bounded subsets of L2M . M

x≥a

0 (ii) the map L2M 3 q 7→ ζ∂kM ∂x m(q) ∈ Cx≥a L2 is analytic, and ζ∂kM ∂x m(q) C 0

x≥a

where K3 can be chosen uniformly on bounded subsets of

L2

≤ K3 kqkL2

M

L2M .

The integral equation for ∂x m(q, x, k) is obtained by taking the derivative in the x-variable of (2.13): +∞ Z e2ik(t−x) q(t) m(q, t, k) dt. (2.28) ∂x m(q, x, k) = − x

Taking the derivative with respect to the k-variable one obtains, for 0 ≤ n ≤ M − 1, ∂kn ∂x m(q, x, k)

+∞ n Z X n =− e2ik(t−x) (2i(t − x))j q(t) ∂kn−j m(q, t, k) dt. j j=0

(2.29)

x

For 0 ≤ j ≤ M introduce the integral operators 0 0 Gj (q) : Cx≥a L2 → Cx≥a L2 ,

q 7→ Gj (q)[f ](x, k) := −

+∞ Z e2ik(t−x) (2i(t − x))j q(t) f (t, k) dt (2.30) x

and rewrite (2.29) in the more compact form ∂kn ∂x m(q)

=

n−1 X j=0

n Gj (q)[∂kn−j m(q)] + Gn (q)[m(q) − 1] + Gn (q)[1]. j

(2.31)

Proposition 2.15 (i) follows from Lemma 2.16 below. The M th derivative requires a separate treatment, as ∂kM m might not be well defined at k = 0. +∞ R 2ik(t−x) Indeed for n = M the integral e q(t) ∂kM m(q, t, k) dt in (2.29) might not be well defined x

near k = 0 since we only know that k∂kM m(q, x, ·) ∈ L2 . To deal with this issue we use the function ζ described above. Multiplying (2.31) with n = M by ζ we formally obtain ζ∂kM ∂x m(q) =

M −1 X j=1

M ζ Gj (q)[∂kM −j m(q)] + ζ GM (q)[m(q) − 1] + ζ GM (q)[1] + G0 (q)[ζ∂kM m(q)]. j

Proposition 2.15 (ii) follows from item (iii) of Lemma 2.16 and the fact that ζ ∈ L∞ : Lemma 2.16. Fix M ∈ Z≥4 and a ∈ R. There exists a constant C > 0 such that (i) for any integer 0 ≤ n ≤ M the following holds: 0 (i1) the map L2M 3 q 7→ Gn (q)[1] ∈ Cx≥a L2 is analytic. Moreover kGn (q)[1]kC 0

x≥a

C kqkL2 . M

84

L2

≤

0 (i2) The map L2M 3 q 7→ Gn (q) ∈ L L2x≥a L2 , Cx≥a L2 is analytic and kGn (q)[f ]kC 0

x≥a

L2

≤ C kqkL2 kf kL2 M

x≥a

L2

.

0 (ii) For any 0 ≤ j ≤ M − 1, the map L2M 3 q 7→ Gj (q) ∈ L Cx≥a L2 is analytic, and kGj (q)[f ]kC 0

x≥a

L2

≤ C kqkL2 kf kC 0 M

x≥a

L2

.

(iii) For any 1 ≤ n ≤ M − 1, 0 ≤ j ≤ n − 1 and ζ : R → R odd smooth monotone function with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1, the following holds: 0 (iii1) the maps L2M 3 q → Gj (q)[∂kn−j m(q)] ∈ Cx≥a L2 and L2M 3 q → Gn (q)[m(q) − 1] ∈ 0 2 Cx≥a L are analytic. Moreover

2 n−j , kGn (q)[m(q) − 1]kC 0 L2 ≤ K20 kqkL2 ,

Gj (q)[∂k m(q)] 0 2 M

x≥a

Cx≥a L

where K20 can be chosen uniformly on bounded subsets of L2M .

0 L2 is analytic and G0 (q)[ζ∂kM m(q)] C 0 (iii2) The map L2M 3 q → G0 (q)[ζ∂kM m(q)] ∈ Cx≥a

x≥a

K30

2 kqkL2 M

where

K30

can be chosen uniformly on bounded subsets of

L2

≤

L2M .

Proof. As before it’s enough to prove the continuity in q of the maps considered to conclude that they are analytic. 2

(i1) For x ≥ a and any 0 ≤ n ≤ M one has kGn (q)[1](x, ·)kL2 ≤ C

+∞ R x

2

|t−x|2n |q(t)|2 dt ≤ C kqkL2 . M

The claim follows by taking the supremum over x ≥ a in the inequality above. (i2) For x ≥ a and 0 ≤ n ≤ M one has the bound kGn (q)[f ]kC 0 L2 ≤ C kqkL2n kf kL2 L2 , which x≥a x≥a implies the claimed estimate. (ii) For x ≥ a and 0 ≤ j ≤ M − 1 one has the bound kGj (q)[f ]kC 0

x≥a

L2

≤ C kqkL1

M −1

kf kC 0

x≥a

L2

≤ C kqkL2 kf kC 0 M

x≥a

L2

.

(iii1) By Proposition 2.10 one has that for any 1 ≤ n ≤ M − 1 and 0 ≤ j ≤ n − 1 the map 0 L2 is analytic. Since composition of analytic maps is again an L2M 3 q 7→ ∂kn−j m(q) ∈ Cx≥a analytic map, the claim regarding the analyticity follows. The first estimate follows from item (ii). A similar argument can be used to prove the second estimate. 0 (iii2) By Proposition 2.13, the map L2M 3 q 7→ ζ∂kM m(q) ∈ Cx≥a L2 is analytic, implying

regarding the analyticity. The estimate follows from G0 [ζ∂kM m(q)] C 0 L2 ≤ kqkL2 x≥a

M

the claim

M

ζ∂ m(q) 0 k C

The following corollary follows from the results obtained so far: Corollary 2.17. Fix M ∈ Z≥4 . Then the normalized Jost functions mj (q, x, k), j = 1, 2, satisfy: 85

x≥a

L2

.

(i) the maps L2M 3 q 7→ mj (q, 0, ·) − 1 ∈ L2 and L2M 3 q 7→ k α ∂kn mj (q, 0, ·) ∈ L2 are analytic for 1 ≤ n ≤ M − 1 [1 ≤ n ≤ M ] if α = 0 [α = 1]. Moreover kmj (q, 0, ·) − 1kL2 , kk α ∂kn mj (q, 0, ·)kL2 ≤ K1 kqkL2 , M

where K1 > 0 can be chosen uniformly on bounded subsets of

L2M .

(ii) For 0 ≤ n ≤ M −1, the maps L2M 3 q 7→ ∂kn ∂x mj (q, 0, ·) ∈ L2 and L2M 3 q 7→ ζ∂kM ∂x mj (q, 0, ·) ∈ L2 are analytic. Moreover

k∂kn ∂x mj (q, 0, ·)kL2 , ζ∂kM ∂x mj (q, 0, ·) L2 ≤ K2 kqkL2 , M

where K2 > 0 can be chosen uniformly on bounded subsets of

L2M .

Proof. The Corollary follows by evaluating formulas (2.13), (2.19), (2.29) at x = 0 and using the results of Proposition 2.5, 2.10, 2.13 and 2.15.

3

One smoothing properties of the scattering map.

The aim of this section is to prove the part of Theorem 2.1 related to the direct problem. To begin, note that by Theorem 2.4, for q ∈ L24 real valued one has m1 (q, x, k) = m1 (q, x, −k) and m2 (q, x, k) = m2 (q, x, −k); hence S(q, k) = S(q, −k) ,

W (q, k) = W (q, −k) .

(2.32)

Moreover one has for any q ∈ L24 W (q, k)W (q, −k) = 4k 2 + S(q, k)S(q, −k)

∀ k ∈ R \ {0}

(2.33)

which by continuity holds for k = 0 as well. In the case where q ∈ Q, the latter identity implies that S(q, 0) 6= 0. Recall that for q ∈ L24 the Jost solutions f1 (q, x, k) and f2 (q, x, k) satisfy the following integral equations f1 (x, k) = e

ikx

+∞ Z

+

sin k(t − x) q(t)f1 (t, k)dt , k

(2.34)

x

f2 (x, k) = e

−ikx

Zx +

sin k(x − t) q(t)f2 (t, k)dt . k

(2.35)

−∞

Substituting (2.34) and (2.35) into (2.4), (2.3), one verifies that S(q, k), W (q, k) satisfy for k ∈ R and q ∈ L24 S(q, k) =

+∞ Z eikt q(t)f1 (q, t, k)dt ,

(2.36)

−∞ +∞ Z W (q, k) = 2ik − e−ikt q(t)f1 (q, t, k)dt . −∞

86

(2.37)

Note that the integrals above are well defined thanks to the estimate in item (ii) of Theorem 2.4. Inserting formula (2.34) into (2.36), one gets that 1 k

S(q, k) = F− (q, k) + O

.

The main result of this section is an estimate of A(q, k) := S(q, k) − F− (q, k) ,

(2.38)

saying that A is 1-smoothing. To formulate the result in a precise way, we need to introduce the following Banach spaces for M ∈ Z≥1 H∗M := {f ∈ HCM −1 :

f (k) = f (−k),

k∂kM f ∈ L2 } ,

HζM := {f ∈ HCM −1 :

f (k) = f (−k),

ζ∂kM f ∈ L2 } ,

where ζ : R → R is an odd monotone C ∞ function with ζ(k) = k for |k| ≤ 1/2 and ζ(k) = 1 for k ≥ 1. The norms on H∗M and HζM are given by

2 2 2 kf kH M := kf kH M −1 + k∂kM f L2 , ∗

H∗M

2 2 2 kf kH M := kf kH M −1 + ζ∂kM f L2 . ζ

C

C

HζM

and are real Banach spaces. We will use also the complexification of the Banach Note that spaces above, in which the reality condition f (k) = f (−k) is dropped: M H∗,C := {f ∈ HCM −1 :

M Hζ,C := {f ∈ HCM −1 :

k∂kM f ∈ L2 },

ζ∂kM f ∈ L2 }.

Note that for any M ≥ 2 M M M (i) HCM ⊂ Hζ,C and H∗,C ⊂ Hζ,C ,

M (ii) f g ∈ Hζ,C

We can now state the main theorem of this section. Let L2M,R

M M ∀ f ∈ H∗,C , g ∈ Hζ,C . (2.39) := f ∈ L2M | f real valued .

Theorem 2.18. Let N ∈ Z≥0 and M ∈ Z≥4 . Then one has: M . (i) The map q 7→ A(q, ·) is analytic as a map from L2M to Hζ,C

(ii) The map q 7→ A(q, ·) is analytic as a map from HCN ∩ L24 to L2N +1 . Moreover 2

kA(q, ·)kL2

N +1

≤ CA kqkH N ∩L2 C

4

where the constant CA > 0 can be chosen uniformly on bounded subsets of HCN ∩ L24 . Furthermore for q ∈ L24,R the map A(q, ·) satisfies A(q, k) = A(q, −k) for every k ∈ R. Thus its restrictions A : L2M,R → HζM and A : H N ∩ L24 → L2N +1 are real analytic. The following corollary follows immediately from identity (2.38), item (ii) of Theorem 2.18 and the properties of the Fourier transform: Corollary 2.19. Let N ∈ Z≥0 . Then the map q 7→ S(q, ·) is analytic as a map from HCN ∩ L24 to L2N . Moreover kS(q, ·)kL2 ≤ CS kqkH N ∩L2 N

C

4

where the constant CS > 0 can be chosen uniformly on bounded subsets of HCN ∩ L24 . 87

In [KST13], it is shown that in the periodic setup, the Birkhoff map of KdV is 1-smoothing. As the map q 7→ S(q, ·) on the spaces considered can be viewed as a version of the Birkhoff map in the scattering setup of KdV, Theorem 2.18 confirms that a result analogous to the one on the circle holds also on the line. The proof of Theorem 2.18 consists of several steps. We begin by proving item (i). Since F− : L2M → HCM is bounded, item (i) will follow from the following proposition: M Proposition 2.20. Let M ∈ Z≥4 , then the map L2M 3 q 7→ S(q, ·) ∈ Hζ,C is analytic and

kS(q, ·)kH M ≤ KS kqkL2 , M

ζ,C

where KS > 0 can be chosen uniformly on bounded subsets of L2M . Proof. Recall that f1 (q, x, k) = eikx m1 (q, x, k) and f2 (q, x, k) = e−ikx m2 (q, x, k). The x-independence of S(q, k) implies that S(q, k) = [m1 (q, 0, k), m2 (q, 0, −k)] . (2.40) M M As by Corollary 2.17, mj (q, 0, ·) − 1 ∈ H∗,C and ∂x mj (q, 0, ·) ∈ Hζ,C , j = 1, 2, the identity (2.40) yields

S(q, k) =(m1 (q, 0, k) − 1) ∂x m2 (q, 0, −k) − (m2 (q, 0, −k) − 1) ∂x m1 (q, 0, k) + ∂x m2 (q, 0, −k) − ∂x m1 (q, 0, k) , M thus S(q, ·) ∈ Hζ,C by (2.39). The estimate on the norm kS(q, ·)kH M follows by Corollary 2.17. ζ,C

Proof of Theorem 2.18 (i). The claim is a direct consequence of Proposition 2.20 and the fact that for any real valued potential q, S(q, k) = S(q, −k), F− (q, k) = F− (q, −k) and hence A(q, k) = A(q, −k) for any k ∈ R. In order to prove the second item of Theorem 2.18, we expand the map q 7→ A(q) as a power series of q. More precisely, iterate formula (2.34) and insert the formal expansion obtained in this way in the integral term of (2.36), to get S(q, k) = F− (q, k) +

X sn (q, k) kn

(2.41)

n≥1

where, with dt = dt0 · · · dtn , Z sn (q, k) := ∆n+1

eikt0 q(t0 )

n Y

q(tj ) sin k(tj − tj−1 ) eiktn dt

(2.42)

j=1

is a polynomial of degree n + 1 in q (cf Appendix B) and ∆n+1 is given by ∆n+1 := (t0 , · · · , tn ) ∈ Rn+1 : t0 ≤ · · · ≤ tn . Since by Proposition 2.20 S(q, ·) is in L2 , it remains to control the decay of A(q, ·) in k at infinity. Introduce a cut off function χ with χ(k) = 0 for |k| ≤ 1 and χ(k) = 1 for |k| > 2 and consider the series X χ(k)sn (q, k) . (2.43) χ(k)S(q, k) = χ(k)F− (q, k) + kn n≥1

88

Item (ii) of Theorem 2.18 follows once we show that each term χ(k)sknn(q,k) of the series is bounded as a map from HCN ∩ L24 into L2N +1 and the series has an infinite radius of convergence in L2N +1 . Indeed the analyticity of the map then follows from general properties of analytic maps in complex Banach spaces, see Remark 2.46. In order to estimate the terms of the series, we need estimates on the maps k 7→ sn (q, k). A first trivial bound is given by n+1 1 kqkL1 . (2.44) ksn (q, ·)kL∞ ≤ (n+1)! However, in order to prove convergence of (2.43), one needs more refined estimates of the norm of k 7→ sn (q, k) in L2N . In order to derive such estimates, we begin with a preliminary lemma about oscillatory integrals: Lemma 2.21. Let f ∈ L1 (Rn , C) ∩ L2 (Rn , C). Let α ∈ Rn , α 6= 0 and Z g : R → C, g(k) := eikα·t f (t) dt. Rn

Then g ∈ L2 and for any component αi 6= 0 one has Z kgkL2 ≤

+∞ Z 1/2 ci . . . dtn . |f (t)|2 dti dt1 . . . dt

Rn−1

(2.45)

−∞

Proof. The lemma is a variant of Parseval’s theorem for the Fourier transform; indeed Z Z 2 eikα·(t−s) f (t)f (s) dt ds dk. kgkL2 = g(k) g(k) dk = R

(2.46)

R×Rn ×Rn

R Integrating first in the k variable and using the distributional identity R eikx dk = denotes the Dirac delta function, one gets Z 1 2 f (t) f (s) δ(α · (t − s)) dt ds kgkL2 = 2π

1 2π δ0 ,

where δ0

(2.47)

Rn ×Rn

Choose an index i such that αi 6= 0; then α · (t − s) = 0 implies that si = ti + ci /αi , where P c ci · · · dsn , one has, ci = ˜i = ds1 · · · ds j6=i αj (tj − sj ). Denoting dσi = dt1 · · · dti · · · dtn and dσ integrating first in the variables si and ti , Z Z 1 2 dσi dσ˜i f (t1 , . . . , ti , . . . , tn )f (s1 , . . . , ti + ci /αi , . . . , sn )dti kgkL2 = 2π R Rn−1 ×Rn−1

+∞ +∞ Z 1/2 Z 1/2 2 dσi dσ˜i |f (t)| dti · |f (s)|2 dsi

Z ≤

−∞

Rn−1 ×Rn−1

≤

Z Rn−1

−∞

+∞ Z 1/2 2 d˜ σi |f (s)|2 dsi −∞

89

(2.48)

where in the second line we have used the Cauchy-Schwarz inequality and the invariance of the +∞ R integral |f (s1 , . . . , ti + ci /αi , . . . , sn )|2 by translation. −∞

To get bounds on the norm of the polynomials k 7→ sn (q, k) in L2N it is convenient to study the multilinear maps associated with them: sñ : HCN ∩ L1

n+1

→ L2N , Z

(f0 , · · · , fn ) 7→ sñ (f0 , · · · , fn ) :=

eikt0 f0 (t0 )

∆n+1

n Y

fj (tj ) sin(k(tj − tj−1 )) eiktn dt .

j=1

The boundedness of these multilinear maps is given by the following Lemma 2.22. For each n ≥ 1 and N ∈ Z≥0 , sñ : (HCN ∩ L1 )n+1 → L2N is bounded. In particular there exist constants Cn,N > 0 such that k˜ sn (f0 , . . . , fn )kL2 ≤ Cn,N kf0 kH N ∩L1 · · · kfn kH N ∩L1 . N

(2.49)

C

C

For the proof, introduce the operators Ij : L1 → L∞ , j = 1, 2, defined by I1 (f )(t) :=

+∞ Z f (s) ds

Zt I2 (f )(t) :=

f (s) ds.

(2.50)

−∞

t

It is easy to prove that if u, v ∈ HCN ∩ L1 , then u Ij (v) ∈ HCN ∩ L1 and the estimate ku Ij (v)kH N ≤ C kukH N ∩L1 kvkH N ∩L1 holds for j = 1, 2. C C Q n iktn Proof of Lemma 2.22. As sin x = (eix − e−ix )/2i we can write eikt0 j=1 sin k(tj − tj−1 ) e as a sum of complex exponentials. Note that the arguments of the exponentials are obtained by taking all the possible combinations of ± in the expression t0 ± (t1 − t0 ) ± . . . ± (tn − tn−1 ) + tn . To handle this combinations, define the set n o Λn := σ = (σj )1≤j≤n : σj ∈ {±1} (2.51) and introduce δσ := #{1 ≤ j ≤ n : σj = −1}. For any σ ∈ Λn , define ασ = (αj )0≤j≤n as α0 = (1 − σ1 ),

αj = σj − σj+1 for 1 ≤ j ≤ n − 1, αn = 1 + σn . Pn Note that for any t = (t0 , . . . , tn ), one has ασ · t = t0 + j=1 σj (tj − tj−1 ) + tn . For every σ ∈ Λn , ασ satisfies the following properties: (i) α0 , αn ∈ {2, 0} , αj ∈ {0, ±2} ∀1 ≤ j ≤ n − 1;

(ii) # {j|αj 6= 0} is odd.

(2.52)

Property (i) is obviously true; we prove now (ii) by induction. For n = 1, property (ii) is trivial. To prove the induction step n n + 1, let α0 = 1 − σ1 , . . . , αn = σn − σn+1 , αn+1 = 1 + σn+1 , and 90

define α ˜ n := 1 + σn ∈ {0, 2}. By the induction hypothesis the vector α ˜ σ = (α0 , . . . , αn−1 , α ˜ n ) has an odd number of elements non zero. Case α ˜ n = 0: in this case the vector (α0 , . . . , αn−1 ) has an odd number of non zero elements. Then, since αn = σn − σn+1 = α ˜ n − αn+1 = −αn+1 , one has that (αn , αn+1 ) ∈ {(0, 0), (−2, 2)}. Therefore the vector ασ has an odd number of non zero elements. Case α ˜ n = 2: in this case the vector (α0 , . . . , αn−1 ) has an even number of non zero elements. As αn = 2 − αn+1 , it follows that (αn , αn+1 ) ∈ {(2, 0), (0, 2)}. Therefore the vector ασ has an odd number of non zero elements. This proves (2.52). As n Y X (−1)δσ eikα·t eikt0 sin k(tj − tj−1 ) eiktn = n (2i) j=1 σ∈Λn

P sñ can be written as a sum of complex exponentials, sñ (f0 , . . . , fn )(k) = σ∈Λn where Z sñ,σ (f0 , . . . , fn )(k) = eikα·t f0 (t0 ) · · · fn (tn )dt.

(−1)δσ (2i)n

sñ,σ (f0 , . . . , fn )(k) (2.53)

∆n+1

The case N = 0 follows directly from Lemma 2.21, since for each σ ∈ Λ Qn one has by (2.52) that there exists m with αm 6= 0 implying k˜ sn,σ (f0 , . . . , fn )kL2 ≤ C kfm kL2 j6=m kfj kL1 , which leads to (2.49). We now prove by induction that sñ : (HCN ∩ L1 )n+1 → L2N for any N ≥ 1. We start with n = 1. Since we have already proved that s˜1 is a bounded map from (L2 ∩ L1 )2 to L2 , it is enough to establish the stated decay at ∞. One verifies that 1 s˜1 (f0 , f1 ) = 2i

+∞ +∞ Z Z 1 2ikt e f0 (t) I1 (f1 )(t) dt − e2ikt f1 (t) I2 (f0 )(t) dt 2i −∞

−∞

1 1 = F− (f0 I1 (f1 )) − F− (f1 I2 (f0 )). 2i 2i Hence, for each N ∈ Z≥0 , (f0 , f1 ) 7→ s˜1 (f0 , f1 ) is bounded as a map from (HCN ∩ L1 )2 to L2N . Moreover k˜ s1 (f0 , f1 )kL2 ≤ C1 kf0 I1 (f1 )kH N + kf1 I2 (f0 )kH N ≤ C1,N kf0 kH N ∩L1 kf1 kH N ∩L1 . N

C

C

C

C

We prove the induction step n n + 1 with n ≥ 1 for any N ≥ 1 (the case N = 0 has been already treated). The term sñ+1 (f0 , . . . , fn+1 ) equals Z n Y eikt0 f0 (t0 ) sin k(tj − tj−1 )fj (tj ) eiktn sin k(tn+1 − tn )eik(tn+1 −tn ) fn+1 (tn+1 ) dt ∆n+2

j=1

where we multiplied and divided by the factor eiktn . Writing sin k(tn+1 − tn ) = (eik(tn+1 −tn ) − e−ik(tn+1 −tn ) )/2i , the integral term

+∞ R

eik(tn+1 −tn ) sin k(tn+1 − tn ) fn+1 (tn+1 ) dtn+1 equals

tn

1 2i

+∞ Z 1 e2ik(tn+1 −tn ) fn+1 (tn+1 ) dtn+1 − I1 (fn+1 )(tn ). 2i tn

91

(j)

(j)

Since fn+1 ∈ HCN , for 0 ≤ j ≤ N − 1 one gets fn+1 → 0 when x → ∞, where we wrote fn+1 ≡ ∂kj fn+1 . Integrating by parts N -times in the integral expression displayed above one has +∞ Z N −1 (−1)N 1 X (−1)j+1 (j) 1 (N ) fn+1 (tn ) + e2ik(tn+1 −tn ) fn+1 (tn+1 ) dtn+1 − I1 (fn+1 )(tn ). j+1 N 2i j=0 (2ik) 2i(2ik) 2i tn

Inserting the formula above in the expression for sñ+1 , and using the multilinearity of sñ+1 one gets sñ+1 (f0 , . . . , fn+1 ) = +

(−1)N 2i(2ik)N

N −1 1 X (−1)j+1 1 (j) sñ (f0 , . . . , fn · fn+1 ) − sñ (f0 , . . . , fn I1 (fn+1 )) 2i j=0 (2ik)j+1 2i

Z ∆n+2

eikt0 f0 (t0 )

n Y

(N ) sin k(tj − tj−1 ) fj (tj ) e2iktn+1 fn+1 (tn+1 ) dtn+1 .

(2.54)

(2.55)

j=1

We analyze the first term in the r.h.s. of (2.54). For 0 ≤ j ≤ N − 1, the function fn+1 ∈ HCN −j (j) is in L∞ by the Sobolev embedding theorem. Therefore fn · fn+1 ∈ HCN −j ∩ L1 . By the inductive (j) (j) χ assumption applied to N − j, sñ (f0 , . . . , fn · fn+1 ) ∈ L2N −j . Therefore (2ik) ñ (f0 , . . . , fn · fn+1 ) ∈ j+1 s L2N , where χ is chosen as in (2.43). For the second term in (2.54) it is enough to note that fn I1 (fn+1 ) ∈ HCN ∩L1 and by the inductive assumption it follows that sñ (f0 , . . . , fn I1 (fn+1 )) ∈ L2N . We are left with (2.55). Due to the factor (2ik)N in the denominator, we need just to prove that the integral term is L2 integrable in the k-variable. Since the oscillatory factor e2iktn+1 doesn’t get canceled when we express the sine functions with exponentials, we can apply Lemma 2.21, integrating first in L2 w.r. to the variable tn+1 , getting (j)

(N ) kχ · (2.55)kL2 ≤ Cn+1,N fn+1

L2

N

n Y

kfj kL1 .

j=0

Putting all together, it follows that sñ+1 is bounded as a map from (HCN ∩ L1 )n+2 to L2N for each N ∈ Z≥0 and the estimate (2.49) holds. By evaluating the multilinear map sñ on the diagonal, Lemma 2.22 says that for any N ≥ 0, n+1

ksn (q, ·)kL2 ≤ Cn,N kqkH N ∩L1 , N

∀n ≥ 1.

(2.56)

C

Combining the L∞ estimate (2.44) with (2.56) we can now prove item (ii) of Theorem 2.18: Proof of Theorem 2.18 (ii). Let χ be the cut off function introduced in (2.43) and set ˜ k) := A(q,

∞ X χ(k)sn (q, k) . kn n=1

(2.57)

˜ ·) is an absolutely and uniformly convergent series in L2 We now show that for any ρ > 0, A(q, N +1 for q in Bρ (0), where Bρ (0) is the ball in HCN ∩ L1 with center 0 and radius ρ. By (2.56) the map

92

PN +1 q 7→ n=1 χ(k)sknn(q,k) is analytic as a map from HCN ∩L1 to L2N +1 , being a finite sum of polynomials - cf. Remark 2.46. It remains to estimate the sum ˜ k) − AÑ +2 (q, k) := A(q,

N +1 X n=1

χ(k)sn (q, k) . kn

∞

It is absolutely convergent since by the L estimate (2.44)

X χ(k) X kqkn+1

X χsn (q, ·) L1

≤ ks (q, ·)k ≤ C ∞ n L

kn 2

n k (n + 1)!

n≥N +2

2 LN +1 n≥N +2 n≥N +2

(2.58)

LN +1

for an absolute constant C > 0. Therefore the series in (2.57) converges absolutely and uniformly in Bρ (0) for every ρ > 0. The absolute and uniform convergence implies that for any N ≥ 0, ˜ ·) is analytic as a map from H N ∩ L1 to L2 . q 7→ A(q, N +1 C ˜ ·) It remains to show that identity (2.43) holds, i.e., for every q ∈ HCN ∩ L1 one has χA(q, ·) = A(q, in L2N +1 . Indeed, fix q ∈ HCN ∩ L1 and choose ρ such that kqkH N ∩L1 ≤ ρ. Iterate formula (2.34) C N 0 ≥ 1 times and insert the result in (2.36) to get for any k ∈ R \ {0}, 0

N X sn (q, k) S(q, k) = F− (q, k) + + SN 0 +1 (q, k) , kn n=1

where S

N 0 +1

(q, k) :=

Z

1 k

N 0 +1

e

ikt0

q(t0 )

∆N 0 +2

0 NY +1

q(tj ) sin k(tj − tj−1 ) f1 (q, tN 0 +1 , k) dt .

j=1

By the definition (2.38) of A(q, k) and the expression of SN 0 +1 displayed above 0

χ(k)A(q, k) −

N X χ(k)sn (q, k) = χ(k)SN 0 +1 (q, k), kn n=1

∀N 0 ≥ 1 .

Let now N 0 ≥ N , then by Theorem 2.4 (ii) there exists a constant Kρ , which can be chosen uniformly on Bρ (0) such that N 0 +2

kχSN 0 +1 (q, ·)kL2

N +1

≤ Kρ

kqkL1 1

(N 0 + 2)!

0

≤ Kρ

ρN +2 → 0, (N 0 + 2)!

when N 0 → ∞ ,

where for the last inequality we used that kqkL1 ≤ C kqkL2 for some absolute constant C > 0. Since 1 2 PN 0 ˜ k) in L2 , it follows that χ(k)A(q, k) = A(q, ˜ k) in L2 . limN 0 →0 n=1 χ(k)sknn(q,k) = A(q, N +1 N +1 For later use we study regularity and decay properties of the map k 7→ W (q, k). For q ∈ L24 real valued with no bound states it follows that W (q, k) 6= 0, ∀ Im k ≥ 0 by classical results in scattering theory. We define QC := q ∈ L24 : W (q, k) 6= 0, ∀ Im k ≥ 0 , QN,M := QC ∩ HCN ∩ L2M . (2.59) C 93

We will prove in Lemma 2.25 below that QN,M is open in HCN ∩ L2M . Finally consider the Banach C M space WC defined for M ≥ 1 by ∂k f ∈ HCM −1 } ,

WCM := {f ∈ L∞ : 2

2

(2.60)

2

endowed with the norm kf kW M = kf kL∞ + k∂k f kH M −1 . C

C

Note that HCM ⊆ WCM for any M ≥ 1 and M gh ∈ Hζ,C

M ∀ g ∈ Hζ,C , ∀ h ∈ WCM .

(2.61)

The properties of the map W are summarized in the following Proposition: Proposition 2.23. For M ∈ Z≥4 the following holds: M (i) The map L2M 3 q 7→ W (q, ·) − 2ik + F− (q, 0) ∈ Hζ,C is analytic and

kW (q, ·) − 2ik + F− (q, 0)kH M ≤ CW kqkL2 , M

ζ,C

where the constant CW > 0 can be chosen uniformly on bounded subsets of L2M . (ii) The map Q0,M 3 q 7→ 1/W (q, ·) ∈ L∞ is analytic. C (iii) The maps Q0,M 3 q 7→ C

∂kj W (q, ·) ∈ L2 for 0 ≤ j ≤ M − 1 W (q, ·)

and

Q0,M 3 q 7→ C

ζ∂kM W (q, ·) ∈ L2 W (q, ·)

are analytic. Here ζ is a function as in (2.8). Proof. The x-independence of the Wronskian function (2.3) implies that W (q, k) = 2ik m2 (q, 0, k) m1 (q, 0, k) + [m2 (q, 0, k), m1 (q, 0, k)].

(2.62)

Introduce for j = 1, 2 the functions m ` j (q, k) := 2ik (mj (q, 0, k) − 1). By the integral formula (2.13) one verifies that +∞ Z

m ` 1 (q, k) =

e2ikt − 1 q(t) (m1 (q, t, k) − 1) dt +

0

Z0 m ` 2 (q, k) =

+∞ +∞ Z Z e2ikt q(t) dt − q(t) dt; 0

e−2ikt − 1 q(t) (m2 (q, t, k) − 1) dt +

−∞

0

Z0 −∞

e−2ikt q(t) dt −

(2.63)

Z0 q(t) dt. −∞

A simple computation using (2.62) shows that W (q, k) − 2ik + F− (q, 0) = I + II + III where I := m ` 1 (q, k) + m ` 2 (q, k) + F− (q, 0), II := m ` 1 (q, k)(m2 (q, 0, k) − 1)

and

III := [m2 (q, 0, k), m1 (q, 0, k)].

(2.64)

M We prove now that each of the terms I, II and III displayed above is an element of Hζ,C . We begin by discussing the smoothness of the functions k 7→ m ` j (q, k), j = 1, 2. For any 1 ≤ n ≤ M,

∂kn m ` j (q, k) = 2in ∂kn−1 (mj (q, 0, k) − 1) + 2ik ∂kn mj (q, 0, k) . 94

Thus by Corollary 2.17 (i), m ` j (q, ·) ∈ WCM and q → 7 m ` j (q, ·), j = 1, 2, are analytic as maps from L2M M to WC . Consider first the term III in (2.64). By Corollary 2.17, kIII(q, ·)kH M ≤ KIII kqkL2 , M

ζ,C

where KIII > 0 can be chosen uniformly on bounded subsets of L2M . Arguing as in the proof of M M Proposition 2.20, one shows that it is an element of Hζ,C and it is analytic as a map L2M → Hζ,C . M M Next consider the term II. Since m ` 1 (q, ·) is in WC and m2 (q, 0, ·) − 1 is in Hζ,C , it follows by M M (2.61) that their product is in Hζ,C . It is left to the reader to show that L2M → Hζ,C , q 7→ II(q) is analytic and furthermore kII(q, ·)kH M ≤ KII kqkL2 , where KII > 0 can be chosen uniformly on M

ζ,C

bounded subsets of L2M . Finally let us consider term I. By summing the identities for m ` 1 and m ` 2 in equation (2.63), one gets that +∞ +∞ Z Z 2ikt q(t) (m1 (q, t, k) − 1) dt e q(t) m1 (q, t, k) dt − m ` 1 (q, k) + m ` 2 (q, k) + F− (q, 0) = 0

0

Z0 +

e−2ikt q(t) m2 (q, t, k) −

−∞

(2.65)

Z0 q(t) (m2 (q, t, k) − 1) dt. −∞

We study just the first line displayed above, the second being treated analogously. By equation +∞ R 2ikt (2.28) one has that e q(t) m1 (q, t, k) dt = ∂x m(q, 0, k), which by Corollary 2.17 is an element 0 M M . Furthermore, by Proposition 2.10 and Proposition of Hζ,C and analytic as a function L2M → Hζ,C +∞ R M 2.13 it follows that k 7→ q(t) (m1 (q, t, k) − 1) dt is an element of Hζ,C and it is analytic as a 0 M function L2M → Hζ,C . This proves item (i). By Corollary 2.17, it follows that kI(q, ·)kH M ≤ ζ,C

KI kqkL2 , where KI > 0 can be chosen uniformly on bounded subsets of L2M . M

0,4 We prove now item (ii). By the definition of QC , for q ∈ QC the function W (q, k) 6= 0 for any k with Im k ≥ 0. By Proposition 2.20 (ii) and the condition M ≥ 4, it follows that W (q, k) = 2ik + L∞ ; therefore the map Q0,M 3 q 7→ 1/W (q) ∈ L2 is analytic. C Item (iii) follows immediately from item (i) and (ii).

Lemma 2.24. For any q ∈ Q0,4 , W (q, 0) < 0. Proof. Let q ∈ Q0,4 and κ ≥ 0. By formulas (2.34) and (2.35) with k = iκ, it follows that fj (q, x, iκ) (j = 1, 2) is real valued (recall that q is real valued). By the definition W (q, iκ) = [f2 , f1 ] (q, iκ) it follows that for κ ≥ 0, W (q, iκ) is real valued. As q is generic, W (q, iκ) has no zeroes for κ ≥ 0. Furthermore for large κ we have W (q, iκ) ∼ 2i(iκ) = −2κ. Thus W (q, iκ) < 0 for κ ≥ 0. We are now able to prove the direct scattering part of Theorem 2.1. Proof of Theorem 2.1: direct scattering part. Let N ≥ 0, M ≥ 4 be fixed integers. First we remark that S(q, ·) is an element of S M,N if q ∈ QN,M . By (2.32), S(q, ·) satisfies (S1). To see that S(q, 0) > 0 recall that S(q, 0) = −W (q, 0), and by Lemma 2.24 W (q, 0) < 0. Thus S(q, ·) satisfies (S2). Finally by Corollary 2.19 and Proposition 2.20 it follows that S(q, ·) ∈ S M,N . The analyticity properties of the map q 7→ S(q, ·) and q 7→ A(q, ·) follow by Corollary 2.19, Proposition 2.20 and 95

Theorem 2.18. We conclude this section with a lemma about the openness of QN,M and S M,N . Lemma 2.25. For any integers N ≥ 0, M ≥ 4, QN,M [QN,M ] is open in H N ∩ L2M [HCN ∩ L2M ]. C Proof. The proof can be found in [KT88]; we sketch it here for the reader’s convenience. By a classical result in scattering theory [DT79], W (q, k) admits an analytic extension to the upper plane Im k ≥ 0. By definition (2.59) one has QC = {q ∈ L24 : W (q, k) 6= 0 ∀ Im k ≥ 0}. Using that (q, k) 7→ W (q, k) is continuous on L24 × R and that by Proposition 2.23, kW (q, ·) − 2ikkL∞ is bounded locally uniformly in q ∈ L24 one sees that QC is open in L24 . The remaining statements follow in a similar fashion. M Denote by Hζ,C the complexification of the Banach space HζM , in which the reality condition f (k) = f (−k) is dropped: M Hζ,C := {f ∈ HCM −1 :

ζ∂kM f ∈ L2 }.

(2.66)

M On Hζ,C ∩ L2N with M ≥ 4, N ≥ 0, define the linear functional M Γ0 : Hζ,C ∩ L2N → C,

h 7→ h(0).

M By the Sobolev embedding theorem Γ0 is a linear analytic map on Hζ,C ∩ L2N . In view of the M,N definition (2.9), S M,N ⊆ HζM . Furthermore denote by SC the complexification of S M,N . It M consists of functions σ : R → C with 0 and σ ∈ Hζ,C ∩ L2N . In the following we denote by C n,γ (R, C), with n ∈ Z≥0 and 0 < γ ≤ 1, the space of complexvalued functions with n continuous derivatives such that the nth derivative is Hölder continuous with exponent γ.

Lemma 2.26. For any integers M ≥ 4, N ≥ 0 the subset S M,N [SCM,N ] is open in HζM ∩ L2N M ∩ L2N ]. [Hζ,C 4 Proof. Clearly Hζ,C ⊆ HC3 , and by the Sobolev embedding theorem HC3 ,→ C 2,γ (R, C) for any 4 0 < γ < 1/2. It follows that σ → σ(0) is a continuous functional on Hζ,C . In view of the definition M,N of S , the claimed statement follows.

4

Inverse scattering map

The aim of this section is to prove the inverse scattering part of Theorem 2.1. More precisely we prove the following theorem. Theorem 2.27. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. Then the scattering map S : QN,M → S M,N is bijective. Its inverse S −1 : S M,N → QN,M is real analytic. −1 The smoothing and analytic properties of B := S −1 − F− claimed in Theorem 2.1 follow now in a straightforward way from Theorem 2.27 and 2.18.

96

Proof of Theorem 2.1: inverse scattering part. By Theorem 2.27, S −1 : S M,N → QN,M is well −1 defined and real analytic. As by definition B = S −1 − F− and S = F− + A one has B ◦ S = −1 −1 Id − F− ◦ S = −F− ◦ A or −1 B = −F− ◦ A ◦ S −1 . Hence, by Theorem 2.18 and Theorem 2.27, for any M ∈ Z≥4 and N ∈ Z≥0 the restriction B : S M,N → H N +1 ∩ L2M −1 is a real analytic map. The rest of the section is devoted to the proof of Theorem 2.27. By the direct scattering part of Theorem 2.1 proved in Section 3, S(QN,M ) ⊆ S M,N . Furthermore, the map S : Q → S is 1-1, see [KT86, Section 4]. Thus also its restriction S|QN,M : QN,M → S M,N is 1-1. Let us denote by H : L2 → L2 the Hilbert transform Z ∞ v(k 0 ) 0 1 dk . (2.67) H(v)(k) := − p. v. 0 π −∞ k − k We collect in Appendix E some well known properties of the Hilbert transform which will be exploited in the following. In order to prove that S : QN,M → S M,N is onto, we need some preparation. Following [KT86] define for σ ∈ S M,N , 1 i 4(k 2 + 1) ω(σ, k) := exp l(σ, k) + H(l(σ, ·))(k) , l(σ, k) := log , k ∈ R (2.68) 2 2 4k 2 + σ(k)σ(−k) and

1 ω(σ, k) := , w(σ, k) 2i(k + i) σ(−k) ρ+ (σ, k) := , w(σ, k)

2ik , w(σ, k) σ(k) ρ− (σ, k) := . w(σ, k) τ (σ, k) :=

(2.69)

The aim is to show that ρ+ (σ, ·), ρ− (σ, ·) and τ (σ, ·) are the scattering data r+ , r− and t of a potential q ∈ QN,M . In the next proposition we discuss the properties of the map σ → l(σ, ·). To this aim we introduce, for M ∈ Z≥2 and ζ as in (2.8), the auxiliary Banach space WζM := {f ∈ L∞ : f (k) = f (−k),

∂kn f ∈ L2 for 1 ≤ n ≤ M − 1 ,

ζ∂kM f ∈ L2 }

(2.70)

and its complexification M Wζ,C := {f ∈ L∞ :

∂kn f ∈ L2 for 1 ≤ n ≤ M − 1 ,

ζ∂kM f ∈ L2 } ,

(2.71)

2 2 2 2 both endowed with the norm kf kW M := kf kL∞ + k∂k f kH M −2 + ζ∂kM f L2 . ζ,C

C

Proposition 2.28. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. The map S M,N → HζM , σ → l(σ, ·) is real analytic. Proof. Denote by h(σ, k) :=

4(k 2 + 1) . 4k 2 + σ(k)σ(−k) 97

We show that the map S M,N → WζM , σ → h(σ, ·) is real analytic. First note that the map SCM,N → L∞ , assigning to σ the function σ(k)σ(−k) is analytic by the Sobolev embedding theorem. For σ ∈ SCM,N write σ = σ1 + iσ2 , where σ1 := 0 such that |l(σ, k)| ≤ C3 /|k|2 , ∀σ ∈ Vσ0 . Thus σ → l(σ, ·) is real analytic as a map from h(σ,·) S M,N to L2 . One verifies that ∂k log(h(σ, ·)) = ∂kh(σ,·) is in L2 and one shows by induction that M,N M the map S → Hζ , σ 7→ l(σ, ·) is real analytic. In the next proposition we discuss the properties of the map σ → ω(σ, ·). Proposition 2.29. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. The map S M,N → WζM , σ → ω(σ, ·) is real analytic. Furthermore ω(σ, ·) has the following properties: (i) ω(σ, k) extends analytically in the upper half plane Im k > 0, and it has no zeroes in Im k ≥ 0. (ii) ω(σ, k) = ω(σ, −k) ∀k ∈ R. (iii) For every k ∈ R ω(σ, k)ω(σ, −k) =

4(k 2 + 1) . 4k 2 + σ(k)σ(−k)

M M Proof. By Lemma 2.60, the Hilbert transform is a bounded linear operator from Hζ,C to Hζ,C . By Proposition 2.28 it then follows that the map

S M,N → HζM ,

σ 7→ H(l(σ, ·))

is real analytic as well. Since the exponential function is real analytic and ∂k ω(σ, ·) = 21 ∂k (l(σ, ·) + iH(l(σ, ·)))ω(σ, ·), one proves by induction that S M,N → WζM , σ → ω(σ, ·) is real analytic. Properties (i)–(iii) are proved in [KT86, Section 4]. 1 w(σ,·) . The following ω(σ,k) 1 w(σ,k) = 2i(k+i) .

Next we consider the map σ → Proposition 2.29 and the definition

98

proposition follows immediately from

Proposition 2.30. The map S M,N → HCM −1 , σ → S M,N → L2 , are real analytic. The function (i)

1 w(σ,k)

=

1 w(σ,−k)

1 w(σ,·)

σ → ∂kn

1 w(σ,·)

is real analytic. Furthermore the maps

2ik , w(σ, ·)

1≤n≤M

fulfills

for every k ∈ R.

2ik (ii) w(σ,k) ≤ 1 for every k ∈ R. (iii) For every k ∈ R

w(σ, k)w(σ, −k) = 4k 2 + σ(k)σ(−k) .

In particular |w(σ, k)| > 0 for every k ∈ R and σ ∈ S M,N . Now we study the properties of ρ+ (σ, ·) and ρ− (σ, ·) defined in formulas (2.69). Proposition 2.31. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. Then the maps S M,N → HζM ∩ L2N , σ → ρ± (σ, ·) are real analytic. There exists C > 0 so that kρ± (σ, ·)kH M ∩L2 ≤ C kσkH M ∩L2 , where ζ,C

N

ζ

C depends locally uniformly on σ ∈ S M,N . Furthermore the following holds:

N

(i) unitarity: τ (σ, k)τ (σ, −k) + ρ± (σ, k)ρ± (σ, −k) = 1 and ρ+ (σ, k)τ (σ, k) + ρ− (σ, k)τ (σ, k) = 0 for every k ∈ R . (ii) reality: τ (σ, k) = τ (σ, −k), ρ± (σ, k) = ρ± (σ, −k); (iii) analyticity: τ (σ, k) admits an analytic extension to {Im k > 0}; (iv) asymptotics: τ (σ, z) = 1 + O(1/|z|) as |z| → ∞, Im z ≥ 0, and ρ± (σ, k) = O(1/k), as |k| → ∞, k real; (v) rate at k = 0: |τ (σ, z)| > 0 for z 6= 0, Im z ≥ 0 and |ρ± (σ, k)| < 1 for k 6= 0. Furthermore τ (σ, z) =αz + o(z), 1 + ρ± (σ, k) =β± k + o(k),

α 6= 0,

Im z ≥ 0

k ∈ R;

Proof. The real analyticity of the maps S M,N → HζM ∩ L2N , σ → ρ± (σ, ·) follows from Proposition 2.30 and the definition ρ± (σ, k) = σ(∓, k)/w(σ, k) (see also the proof of Proposition 2.32). Since 1 σ 7→ w(σ,·) is real analytic, it is locally bounded, i.e., there exists C > 0 so that kρ± (σ, ·)kH M ∩L2 ≤ N

ζ,C

C kσkH M ∩L2 , where C depends locally uniformly on σ ∈ S M,N . Properties (i), (ii), (v) follow now N ζ by simple computations. Property (iii) − (iv) are proved in [KT86, Lemma 4.1]. Finally define the functions R± (σ, k) := 2ikρ± (σ, k) .

(2.74)

Proposition 2.32. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. Then the maps S → ∩ L2N , σ → R± (σ, ·) are real analytic. There exists C > 0 so that kR± (σ, ·)kH M ∩L2 ≤ C kσkH M ∩L2 , M,N

C

N

where C depends locally uniformly on σ ∈ S M,N . Furthermore the following holds: 99

HCM

ζ

N

(i) R± (σ, k) = R± (σ, −k) for every k ∈ R. (ii) |R± (σ, k)| < 2|k| for any k ∈ R \ {0}. σ(∓k) Proof. In order to prove the statements, we will use that R± (σ, k) = 2ik w(σ,k) . We will consider just R− , since the analysis for R+ is identical. To simplify the notation, we will denote R− (σ, ·) ≡ R(σ, ·). By Proposition 2.30(ii), |R(σ, k)| ≤ |σ(k)| , thus R(σ, ·) ∈ L2N . In order to prove that R(σ, ·) ∈ M HC , take n derivatives (1 ≤ n ≤ M ) of R(σ, ·) to get the identity

∂kn R(σ, k) =

n−1 X n j 2ik n−j 2ik 2ik ∂kn σ(k) + ∂k σ(k) + ∂kn σ(k) . ∂k w(σ, k) w(σ, k) w(σ, k) j j=1

(2.75)

We show now that each term of the r.h.s. of the identity above is in L2 . Consider first the term 2ik ∂kn σ(k). If 1 ≤ n < M , then ∂kn σ ∈ L2 and |2ik/w(σ, k)| ≤ 1, thus proving that I1 ∈ L2 . I1 := w(σ,k) If n = M , let χ be a smooth cut-off function with χ(k) ≡ 1 in [−1, 1] and χ(k) ≡ 0 in R \ [−2, 2]. Then one has 2ik 1 χ(k)2ik∂kM σ(k) + (1 − χ(k))∂kM σ(k) . I1 = w(σ, k) w(σ, k) As σ ∈ S M,N it follows that k 7→ χ(k)2ik∂kM σ(k) and k 7→ (1 − χ(k))∂kM σ(k) are in L2 . By 1 2ik Proposition 2.30, w(σ,·) and w(σ,·) are in L∞ . Altogether it follows that I1 ∈ L2 for any 1 ≤ n ≤ M . Pn−1 n j 2ik n−j 2ik Consider now I2 := j=1 j ∂k w(σ,k) ∂k σ(k). By Proposition 2.30, ∂kj w(σ,k) ∈ HC1 2ik ∈ L∞ for every for every 1 ≤ j ≤ M − 1, thus by the Sobolev embedding theorem ∂kj w(σ,k) 1 ≤ j ≤ M − 1. As ∂kn−j σ∈ L2 for 1 ≤ j ≤ n − 1 < M , it followsthat I2 ∈L2 for any 1 ≤ n ≤ M . 2ik 2ik σ(k). By Proposition 2.30, ∂kn w(σ,k) ∈ L2 for any 1 ≤ n ≤ Finally consider I3 := ∂kn w(σ,k) M . Since σ ∈ L∞ , I3 ∈ L2 for any 1 ≤ n ≤ M . Altogether we proved that R(σ, ·) ∈ HCM ∩ L2N . The claimed estimate on kR(σ, ·)kH M ∩L2 and N C item (i) and (ii) follow in a straightforward way. The real analyticity of the map S M,N → HCM ∩L2N , σ → R(σ, ·) follows by Proposition 2.30. For σ ∈ S M,N , define the Fourier transforms −1 F± (σ, y) := F± (ρ± (σ, ·))(y) =

Then ±∂y F± (σ, y) =

1 π

1 π

Z

ρ± (σ, k)e±2iky dk .

(2.76)

R

+∞ Z −1 2ikρ± (σ, k)e±2iky dk = F± (R± (σ, ·))(y) .

(2.77)

−∞

In the next proposition we analyze the properties of the maps σ 7→ F± (σ, ·). Proposition 2.33. Let N ∈ Z≥0 and M ∈ Z≥4 be fixed. Then the following holds true: (i) σ 7→ F± (σ, ·) are real analytic as maps from S 4,0 to H 1 ∩ L23 . Moreover there exists C > 0 so that kF± (σ, ·)kH 1 ∩L2 ≤ C kσkH M , where C depends locally uniformly on σ ∈ S M,N . 3

ζ

100

(ii) σ 7→ F±0 (σ, ·) are real analytic as maps from S M,N to H N ∩L2M . Moreover there exists C 0 > 0 so that F±0 (σ, ·) H N ∩L2 ≤ C 0 kσkH M ∩L2 , where C 0 depends locally uniformly on σ ∈ S M,N . ζ

M

N

Proof. By Proposition 2.31, the map S 4,0 → HC3 ∩ L21 , σ → ρ± (σ, ·) is real analytic. Thus item (i) −1 follows by the properties of the Fourier transform. By Proposition 2.31 (ii), F± (σ, ·) = F± (ρ± ) is real valued. Item (ii) follows from (2.77) and the characterizations −1 R± ∈ HCM ⇐⇒ F± (R± ) ∈ L2M

and

−1 R± ∈ L2N ⇐⇒ F± (R± ) ∈ HCN .

(2.78)

The claimed estimates follow from the properties of the Fourier transform, Proposition 2.31 and Proposition 2.32. We are finally able to prove that there exists a potential q ∈ Q with prescribed scattering coefficient σ ∈ S M,N . More precisely the following theorem holds. Theorem 2.34. Let N ∈ Z≥0 , M ∈ Z≥4 and σ ∈ S M,N be fixed. Then there exists a potential q ∈ Q such that S(q, ·) = σ. Proof. Let ρ± := ρ± (σ, ·) and τ := τ (σ, ·) be given by formula (2.69). Let F± (σ, ·) be defined as in (2.76). By Proposition 2.33 it follows that F± (σ, ·) are absolutely continuous and F±0 (σ, ·) ∈ H N ∩ L2M . As M ≥ 4 it follows that Z ∞ (1 + x2 )|F±0 (σ, x)| dx < ∞ . (2.79) −∞

The main theorem in inverse scattering [Fad64] assures that if (2.79) and item (i)–(v) of Proposition 2.32 hold, then there exists a potential q ∈ Q such that r± (q, ·) = ρ± and t(q, ·) = τ , where r± and t are the reflection respectively transmission coefficients defined in (2.5). From the formulas (2.69) it follows that S(q, ·) = σ. It remains to show that q ∈ QN,M and that the map S −1 : S M,N → QN,M is real analytic. We take here a different approach then [KT86]. In [KT86] the authors show that the map S is complex differentiable and its differential dq S is bounded invertible. Here instead we reconstruct q by solving the Gelfand-Levitan-Marchenko equations and we show that the inverse map S M,N → QN,M , σ 7→ q is real analytic. We outline briefly the procedure. Given two reflection coefficients ρ± satisfying items (i)–(v) of Proposition 2.31 and arbitrary real numbers c+ ≤ c− , it is possible to construct a potential q+ on [c+ , ∞) using ρ+ and a potential q− on (−∞, c− ] using ρ− , such that q+ and q− coincide on the intersection of their domains, i.e., q+ |[c+ ,c− ] = q− |[c+ ,c− ] . Hence q defined on R by q|[c+ ,+∞) = q+ and q|(−∞,c− ] = q− is well defined, q ∈ Q and r± (q, ·) = ρ± , i.e., ρ+ and ρ− are the reflection coefficients of the potential q [Fad64, Mar86, DT79]. We postpone the details of this procedure to the next section.

4.1

Gelfand-Levitan-Marchenko equation

In this section we prove how to construct for any σ ∈ S M,N two potentials q+ and q− with N N q+ ∈ Hx≥c ∩ L2M,x≥c respectively q− ∈ Hx≤c ∩ L2M,x≤c , where for any c ∈ R and 1 ≤ p ≤ ∞ n Lpx≥c := f : [c, +∞) → C : kf kLp

x≥c

101

0 can be chosen locally uniformly in σ ∈ S M,0 . Proof. We prove the result just for R+,c , since for R−,c the proof is analogous. As before, we suppress the subscript ”+” from the various objects. Consider the Gelfand-Levitan-Marchenko equation (2.84). Multiply it by hxiM −3/2 to obtain h i (Id + Kx,σ ) hxiM −3/2 Eσ (x, y) = −hxiM −3/2 Fσ (x + y). (2.100) The function hσ (x, y) := −hxiM −3/2 Fσ (x + y) , 0 0 satisfies hσ (x, ·) ∈ L2y≥0 and one checks that hσ ∈ Cx≥c L2y≥0 ∩ Cx≥c,y≥0 . We show now that 2 2 hσ ∈ Lx≥c Ly≥0 . By Lemma 2.42 (A3) and Proposition 2.33 for N = 0 it follows that

M −3/2 2 hσ 2

hxi

Lx≥c L2y≥0

+∞ Z

2 ≤ Kc hxi2M −2 |Fσ (x)|2 dx ≤ Kc hxiM Fσ0 L2

x≥c

2

≤ Kc kσkH M . ζ,C

c

Consider now hσ (x, 0) = −hxiM −3/2 Fσ (x). By (2.83) it follows that hσ (·, 0) ∈ L2x≥c . Finally the map σ 7→ hσ [σ 7→ hσ (·, 0)] is real analytic as a map from S M,0 to L2x≥c L2y≥0 [L2M −3/2,x≥c ]. 105

Proceeding as in the proof of Lemma 2.51, one shows that there exists a solution Eσ of equa0 0 tion (2.84) which satisfies (i) hxiM −3/2 Eσ ∈ Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 , hxiM −3/2 Eσ (x, ·) ∈ Cy≥0 ,

M −3/2 M −3/2 2 M −3/2 Eσ L2 L2 ≤ Kc kσkH M , (iii) σ 7→ hxi h·i Eσ (·, 0) ∈ Lx≥c , (ii) hxi Eσ [σ 7→ x≥c

ζ,C

y≥0

Eσ (·, 0)] is real analytic as a map from S M,0 to L2x≥c L2y≥0 [L2M −3/2,x≥c ]. Furthermore its derivative ∂x Eσ satisfies the integral equation (Id + Kx,σ ) (∂x Eσ (x, y)) =

−Fσ0 (x

+∞ Z + y) − Fσ0 (x + y + z) Eσ (x, z) dz.

(2.101)

0

˜ σ , where Multiply the equation above by hxiM −3/2 , to obtain (Id + Kσ ) hxiM −3/2 ∂x Eσ = h ˜ σ (x, y) := −hxiM −3/2 h0 (x, y) − h σ

+∞ Z Fσ0 (x + y + z) hxiM −3/2 Eσ (x, z) dz .

(2.102)

0

˜ σ ∈ L2 L2 and σ 7→ h ˜ σ is real analytic as a map where h0σ (x, y) := Fσ0 (x + y). We claim that h x≥c y≥0 M,0 2 2 S → Lx≥c Ly≥0 . By Lemma 2.42 (A0) the first term of (2.102) satisfies

M −1 0

M −3/2 0

hxi hσ 2 ≤ K Fσ L2 ≤ Kc kσkH M ,

hxi c 2 ζ,C Lx≥c Ly≥0

x≥c

and by Lemma 2.42 (A1) the second term of (2.102) satisfies

+∞

Z

M −3/2 2 0 M −3/2 0

F (x + y + z) hxi E (x, z) dz ≤ kF k E ≤ Kc kσkH M .

hxi

2 1 σ σ σ σ L

ζ,C Lx≥c L2y≥0

2 2 0

Lx≥c Ly≥0

˜ σ is real analytic as a map from S M,0 to L2 L2 , being composition of real Moreover σ 7→ h x≥c y≥0 analytic maps.

Thus, by Lemma 2.51, it follows that hxiM −3/2 ∂x Eσ ∈ L2x≥c L2y≥0 , hxiM −3/2 ∂x Eσ L2 L2 ≤ x≥c

Kc kσkH M and σ 7→ h·iM −3/2 ∂x Eσ is real analytic as a map from S M,0 to L2x≥c L2y≥0 . ζ,C Consider now equation (2.84). Evaluate it at y = 0 to get +∞ Z Eσ (x, 0) = −Fσ (x) − Fσ (x + z)Eσ (x, z) dz . 0

Take the x-derivative of the equation above and multiply it by hxiM to obtain M

M

hxi ∂x Eσ (x, 0) = − hxi

Fσ0 (x)

+∞ Z − hxi3/2 Fσ0 (x + z)hxiM −3/2 Eσ (x, z) dz 0

+∞ Z

hxi3/2 Fσ (x + z)hxiM −3/2 ∂x Eσ (x, z) dz .

− 0

106

y≥0

We prove now that ∂x Eσ (·, 0) ∈ L2M,x≥c and σ 7→ ∂x Eσ (·, 0) is real analytic as a map from S M,0 to L2M,x≥c . The result follows by Proposition 2.33 and Lemma 2.42 (A2). Indeed one has that σ 7→ Fσ0 [σ 7→ Fσ ] is real analytic as a map from S M,0 to L2M [L23/2 ], and we proved above that σ 7→ h·iM −3/2 Eσ and σ 7→ h·iM −3/2 ∂x Eσ are real analytic as maps from S M,0 to L2x≥c L2y≥0 . Combining the results of Proposition 2.37 and Proposition 2.39, we can prove Theorem 2.36. Proof of Theorem 2.36. It follows from Proposition 2.33, Proposition 2.37 and Proposition 2.39 by restricting the scattering maps R±,c to the spaces S M,N = S 4,N ∩ S M,0 . Using the results of Theorem 2.36 and Theorem 2.34 we can prove Theorem 2.27, showing that S −1 : S N,M → QN,M is real analytic. Proof of Theorem 2.27. Let σ ∈ S M,N . By Theorem 2.34 there exists q ∈ Q with S(q, ·) = σ. Now let c+ ≤ c ≤ c− be arbitrary real numbers and consider R+,c+ (σ) and R−,c− (σ), where R±,c± are defined in (2.87). By classical inverse scattering theory [Fad64], [Mar86] the following holds: (i) R+,c (σ) = R−,c (σ) , +

x∈[c+ ,c]

−

x∈[c,c− ]

(ii) the potential qc defined by qc := R+,c+ (σ)1[c,∞) + R−,c− (σ)1(−∞,c]

(2.103)

is in Q and satisfies r+ (qc , ·) = ρ+ (σ, ·), r− (qc , ·) = ρ− (σ, ·) and t(qc , ·) = τ (σ, ·). Thus by formulas (2.5) and (2.69) it follows that S(qc , ·) = σ. N Since S is 1-1 it follows that qc ≡ q. Finally, by Theorem 2.36, S M,N → Hx≥c ∩ L2M,x≥c+ , + N σ 7→ R+,c+ (σ) and S M,N → Hx≤c ∩ L2M,x≤c− , σ 7→ R−,c− (σ) are real analytic. It follows that − −1 N 2 q ∈ H ∩ LM and the map S : σ → q is real analytic.

5

Proof of Corollary 2.2 and Theorem 2.3

This section is devoted to the proof of Corollary 2.2 and Theorem 2.3. Both results are easy applications of Theorem 2.1. Proof of Corollary 2.2. Let N ≥ 0, M ≥ 4 be fixed integers. Fix q ∈ QN,M . By Theorem 2.1 the scattering map S(q, ·) is in S M,N . Furthermore by the definition (2.10) of I(q, k) there exists a constant C > 0 such that for any |k| ≥ 1 |I(q, k)| ≤

C|S(q, k)|2 . |k|

In particular I(q, ·) ∈ L12N +1 ([1, ∞), R). By the real analyticity of the map q 7→ S(q, ·), it follows that QN,M → L12N +1 ([1, ∞), R), q 7→ I(q, ·)|[1,∞) is real analytic. Now let us analyze I(q, k) for 0 ≤ k ≤ 1. By the definition (2.10) of I(q, k) one has k 4k 2 k 4(k 2 + 1) I(q, k) + log = − log . π 4(k 2 + 1) π 4k 2 + S(q, k)S(q, −k)

107

4(k2 +1) By Proposition 2.28, the map S M,N → HζM ([0, 1], R), σ → l(σ, k) := log 4k2 +σ(k)σ(−k) is real analytic. Thus also the map QN,M → HζM ([0, 1], R), q → l(S(q), ·) is real analytic, being composition of real analytic maps. It follows that the map q 7→ − πk l(S(q), k) is real analytic as a map from QN,M to H M ([0, 1], R). For t ∈ R and σ ∈ HC1 , let us denote by 3

Ωt (σ)(k) := e−i8k t σ(k) .

(2.104)

We prove the following lemma. Lemma 2.40. Let N, M be integers with N ≥ 2M ≥ 2. Let σ ∈ S M,N . Then Ωt (σ) ∈ S M,N , ∀t ≥ 0. Proof. As a first step we show that Ωt (σ) ∈ S for every t ≥ 0. Since Ωt (σ)(0) = σ(0) > 0 and Ωt (σ)(k) = Ωt (σ)(−k), Ωt (σ) satisfies (S1) and (S2) for every t ≥ 0. Thus Ωt (σ) ∈ S , ∀t ≥ 0. M ∩ L2N . Clearly |Ωt (σ)(k)| ≤ |σ(k)|, thus Ωt (σ) ∈ L2N , ∀t ≥ 0. Now Next we show that Ωt (σ) ∈ Hζ,C t M we show that Ω (σ) ∈ Hζ,C , ∀t ≥ 0. In particular we prove that ζ∂kM Ωt (σ) ∈ L2 , the other cases being analogous. Using the expression (2.104) one gets that   M −1 X 3 M j M ζ(k)∂kM Ωt (σ)(k) = e−i8k t ζ(k)∂kM σ(k) + −i24tk 2 ζ(k)∂kM −j σ(k) + −i24tk 2 ζ(k)σ(k) . j j=1 As σ ∈ S M,N , the first and last term in the r.h.s. above are in L2 . Now we show that for 1 ≤ j ≤ M − 1, |k|2j ζ∂kM −j σ ∈ L2 . We will use the following interpolating estimate, proved in [NP09, Lemma 4]. Assume that J a f := (1 − ∂k2 )a/2 f ∈ L2 and hkib f := (1 + |k|2 )b/2 f ∈ L2 . Then for any θ ∈ (0, 1)

θb (1−θ)a θ 1−θ f 2 ≤ c kf kL2 kf kH a . (2.105)

hki J C b L

HCM

Note that ζσ ∈ ∩ L2N , thus we can apply estimate (2.105) Nj to obtain that hki M ∂kM −j (ζσ) ∈ L2 . Since N ≥ 2M , we have

j with f = ζσ, b = N , a = M , θ = M , hki2j ∂kM −j (ζσ) ∈ L2 . By integration

by parts hki2j ζ(k)∂kM −j σ(k) = hki2j ∂kM −j (ζσ) −

M −j X l=1

M −j hki2j ∂kl ζ(k) ∂kM −j−l σ(k) . l

Since for any l ≥ 1 the function ∂kl ζ has compact support, it follows that the r.h.s. above is in L2 . Thus for every 1 ≤ j ≤ M − 1 we have hki2j ζ(k)∂kM −j σ ∈ L2 and it follows that ζ∂kM Ωt (σ) ∈ L2 for every t ≥ 0. Remark 2.41. One can adapt the proof above, putting ζ(k) ≡ 1, to shows that the spaces H N ∩L2M , with integers N ≥ 2M ≥ 2, are invariant by the Airy flow. Indeed the Fourier transform F− −1 t conjugates the Airy flow with the linear flow Ωt , i.e., UAiry = F− ◦ Ωt ◦ F − .

108

Proof of Theorem 2.3. Recall that by [GGKM74] the scattering map S conjugate the KdV flow 3 3 with the linear flow Ωt (σ)(k) := e−i8π k t σ(k), i.e., t UKdV = S −1 ◦ Ωt ◦ S ,

(2.106)

−1 t whereas UAiry = F− ◦ Ωt ◦ F− . Take now q ∈ QN,M , where N, M are integers with N ≥ 2M ≥ 8. By Theorem 2.1, S(q) ≡ S(q, ·) ∈ S M,N . By Lemma 2.40 the flow Ωt preserves the space S M,N for every t ≥ 0. Thus Ωt ◦ S(q) ∈ S M,N , ∀t ≥ 0. By the bijectivity of S it follows that S −1 ◦ Ωt ◦ S(q) ∈ QN,M ∀t ≥ 0. Thus item (i) is proved. t We prove now item (ii). Remark that by item (i), UKdV (q) ∈ L2M for any t ≥ 0. Since t N 2 UAiry preserves the space H ∩ LM (N ≥ 2M ≥ 8), it follows that for q ∈ QN,M the difference t t UKdV (q) − UAiry (q) ∈ H N ∩ L2M , ∀t ≥ 0. We prove now the smoothing property of the difference −1 t t UKdV (q) − UAiry (q). Since S −1 = F− + B, −1 t UKdV (q) =F− ◦ Ωt ◦ S(q) + B ◦ Ωt ◦ S(q)

(2.107)

and since S = F− + A, −1 −1 −1 ◦ Ωt ◦ A(q) . ◦ Ωt ◦ F− (q) + F− ◦ Ωt ◦ S(q) = F− F−

Hence −1 t t UKdV (q) =UAiry (q) + F− ◦ Ωt ◦ A(q) + B ◦ Ωt ◦ S(q).

(2.108)

t t (q) follows now from the smoothing The 1-smoothing property of the difference UKdV (q) − UAiry properties of A and B described in item (ii) of Theorem 2.1. The real analyticity of the map t t q 7→ UKdV (q) − UAiry (q) follows from formula (2.108) and the real analyticity of the maps A, B and S.

A

Auxiliary results.

For the convenience of the reader in this appendix we collect various known estimates used throughout the paper. Lemma 2.42. Fix an arbitrary real number c. Then the following holds: (A0) The linear map T0 : L21/2,x≥c → L2x≥c L2y≥0 defined by g 7→ T0 (g)(x, y) := g(x + y)

(2.109)

is continuous, and there exists a constant Kc > 0, depending on c, such that kT0 (g)kL2

x≥c

L2y≥0

≤ Kc kgkL2

.

(2.110)

1/2,x≥c

(A1) The bilinear map T1 : L2x≥c × L2x≥c → L2x≥c L2y≥0 defined by (g, h) 7→ T1 (g, h)(x, y) := g(x + y)h(x)

(2.111)

kT1 (g, h)kL2

(2.112)

is continuous, and x≥c

L2y≥0

109

≤ kgkL2

x≥c

khkL2

x≥c

.

(A2) The bilinear map T2 : L2x≥c × L2x≥c L2y≥0 → L2x≥c defined by +∞ Z g(x + z)h(x, z) dz (g, h) 7→ T2 (g, h)(x) :=

(2.113)

0

is continuous, and there exists a constant Kc > 0, depending on c, such that kT2 (g, h)kL2

x≥c

≤ Kc kgkL2

x≥c

khkL2

x≥c

L2y≥0

.

(2.114)

(A3) (Hardy inequality) The linear map T3 : L2m+1,x≥c → L2m,x≥c defined by +∞ Z g→ 7 T3 (g)(x) := g(z)dz x

is continuous, and there exists a constant Kc > 0, depending on c, such that kT3 (g)kL2

m,x≥c

≤ Kc kgkL2

.

m+1,x≥c

(A4) The bilinear map T4 : L1x≥c × L2x≥c L2y≥0 → L2x≥c L2y≥0 defined by +∞ Z (g, h) 7→ T4 (g, h)(x, y) := g(x + y + z)h(x, z)dz

(2.115)

0

is continuous, and there exists a constant Kc > 0, depending on c, such that kT4 (g, h)kL2

x≥c

L2y≥0

≤ Kc kgkL1

x≥c

khkL2

x≥c

L2y≥0

.

(2.116)

(A5) The bilinear map T5 : L2x≥c × L21,x≥c → L2x≥c L2y≥0 defined by +∞ Z (g, h) 7→ T5 (g, h)(x, y) := g(x + y + z)h(x + z)dz

(2.117)

0

is bounded and satisfies kT5 (g, h)kL2

x≥c

L2y≥0

≤ Kc kgkL2

x≥c

khkL2

.

(2.118)

1,x≥c

Proof. Inequality (A1), (A4) can be verified in a straightforward way. To prove (A0) make the change of variable ξ = x + y and remark that +∞ Z +∞ +∞ Z Z 2 |g(x + y)| dx dy ≤ Kc |ξ − c| |g(ξ)|2 dξ . c

0

0

110

We prove now (A2): using Cauchy-Schwartz, one gets

2

+∞

Z

g(x + z)h(x, z) dz

2

0

   +∞  +∞ +∞ Z Z Z 2  |h(x, z)|2 dz  dx ≤ kgkL2 |g(z)|2 dz   ≤

x≥c

x

c

Lx≥c

2

khkL2

x≥c

L2y≥0

0

In order to prove (A3) take a function h ∈ L2x≥c and remark that +∞ +∞ +∞ +∞ Z Z Z z Z Z z Z m m m ˜ |h(x)| dx dz hzi |g(z)| hxi h(x)dx ≤ Kc dz g(z) g(z) dz = dx h(x) hxi c c c

c

x

c

+∞ Rz Z

|h(x)|dx m+1 ≤ Kc dz hzi |g(z)| c ≤ Kc hzim+1 g L2 khkL2 x≥c x≥c |z − c| c

where for the last inequality we used the Hardy-Littlewood inequality. To prove (A4) take a function f ∈ L2x≥c L2y≥0 , define Ωc = [c, ∞) × R+ × R+ and remark that Z |g(x + y + z)| |h(x, z)| |f (x, y)| dx dy dz ≤ Ωc

≤

Z

|g(x + y + z)| |h(x, z)|2 dx dy dz

1/2 Z

Ωc

|g(x + y + z)| |f (x, y)|2 dx dy dz

1/2

Ωc

≤ kgkL1

x≥c

khkL2

x≥c,z≥0

kf kL2

L2 x≥c y≥0

,

where the first inequality follows by writing |g| = |g|1/2 · |g|1/2 and applying Cauchy-Schwartz. To prove (A5) note that

+∞

+∞

Z

Z

≤ kgk |h(z)| dz . g(x + y + z)h(x + z) dz L2x≥c

2 0

+∞

R

By (A3) one has that |h(z)| dz

x

B

x

Ly≥0

≤ Kc khxihkL2

, then (A5) follows.

x≥c

L2x≥c

Analytic maps in complex Banach spaces

In this appendix we recall the definition of an analytic map from [Muj86]. Let E and F be complex Banach spaces. A map P˜ k : E k → F is said to be k-multilinear if P˜ k (u1 , . . . , uk ) is linear in each variable uj ; a multilinear map is said to be bounded if there exist a constant C such that

˜k 1

P (u , · · · , uk ) ≤ C u1 · · · uk ∀u1 , . . . , uk ∈ E.

111

.

Its norm is defined by

˜k

P :=

sup uj ∈E, kuj k≤1

˜k 1

P (u , . . . , uk ) .

A map P k : E → F is said to be a polynomial of order k if there exists a k-multilinear map P˜ k : E → F such that P k (u) = P˜ k (u, . . . , u) ∀u ∈ E. The polynomial is bounded if it has finite norm

k

P := sup P k (u) . kuk≤1

We denote with P k (E, F ) the vector space of all bounded polynomials of order k from E into F . Definition 2.43. Let E and F be complex Banach spaces. Let U be a open subset of E. A mapping f : U → F is said to be analytic if for each a ∈ U there exists a ball Br (a) ⊂ U with center a and radius r > 0 and a sequence of polynomials P k ∈ P k (E, F ), k ≥ 0, such that f (u) =

∞ X

P k (u − a)

k=0

is convergent uniformly for u ∈ Br (a); i.e., for any > 0 there exists K > 0 so that

K

X

k P (u − a) ≤

f (u) −

k=0

for any u ∈ Br (a). Finally let us recall the notion of real analytic map. Definition 2.44. Let E, F be real Banach spaces and denote by EC and FC their complexifications. Let U ⊂ E be open. A map f : U → F is called real analytic on U if for each point u ∈ U there exists a neighborhood V of u in EC and an analytic map g : V → FC such that f = g on U ∩ V . Remark 2.45. The notion of an analytic map in Definition 2.43 is equivalent to the notion of a C-differentiable map. Recall that a map f : U → F , where U , E and F are given as in Definition 2.43, is said to be C-differentiable if for each point a ∈ U there exists a linear, bounded operator A : E → F such that kf (u) − f (a) − A(u − a)kF = 0. lim u→a ku − akE Therefore analytic maps inherit the properties of C-differentiable maps; in particular the composition of analytic maps is analytic. For a proof of the equivalence of the two notions see [Muj86], Theorem 14.7. P∞ m Remark 2.46. Any P k ∈ P k (E, F ) is an analytic map. Let f (u) = m=0 P (u) be a power m m series from E into F with infinite radius of convergence with P ∈ P (E, F ). Then f is analytic ([Muj86], example 5.3, 5.4).

112

C

Properties of the solutions of integral equation (2.93)

In this section we discuss some properties of the solution of equation (2.93) which we rewrite as +∞ Z g(x, y) + Fσ (x + y + z) g(x, z) dz = hσ (x, y) .

(2.119)

0

Here σ ∈ S 4,N , N ≥ 0, hσ is a function hσ : [c, +∞) × [0, +∞) → R, with c arbitrary, which satisfies (P ). We denote by khk0 := khkL2

x≥c

L2y≥0

+ kh(·, 0)kL2

.

(2.120)

x≥c

Furthermore Fσ : R → R is a function that satisfies (H) The map σ 7→ Fσ [σ 7→ Fσ0 ] is real analytic as a map from S 4,N to H 1 ∩ L23 [L24 ]. Moreover the operators Id ± Kx,σ : L2y≥0 → L2y≥0 with Kx,σ defined as +∞ Z Kx,σ [f ](y) := Fσ (x + y + z) f (z) dz

(2.121)

0

are invertible for any x ≥ c, and there exists a constant Cσ > 0, depending locally uniformly 4 on σ ∈ Hζ,C ∩ L2N , such that

sup (Id ± Kx,σ )−1 L(L2 ) ≤ Cσ . (2.122) y≥0

x≥c

Finally σ 7→ (Id ± Kx,σ )−1 are real analytic as maps from S 4,N to L(L2x≥c L2y≥0 ). Remark 2.47. The pairing L(L2x≥c L2y≥0 ) × L2x≥c L2y≥0 → L2x≥c L2y≥0 ,

(H, f ) 7→ H[f ]

is a bounded bilinear map and hence analytic. Let now σ 7→ hσ be a real analytic map from S 4,0 to −1 L2x≥c L2y≥0 and let Kσ as in (H). Then by Lemma 2.54 (iii) it follows that σ 7→ (Id + Kσ ) [hσ ] is real analytic as a map from S 4,0 to L2x≥c L2y≥0 as well. Remark 2.48. By the Sobolev embedding theorem, assumption (H) implies that Fσ ∈ C 0,γ (R, C), γ < 21 . By assumption (H) the map (c, ∞) → L(L2y≥0 ), x 7→ Kx,σ is differentiable and its derivative is the operator +∞ Z 0 Kx,σ [f ](y) = Fσ0 (x + y + z) f (z) dz , (2.123) 0

as one verifies using that for x > c and 6= 0 sufficiently small +∞

Z

Kx+,σ − Kx,σ

Fσ (z + ) − Fσ (z) 0 0 dz

− K − F (z) ≤ x,σ σ

2 L(Ly≥0 )

x

+∞ +∞ Z Z Z 1 0 0 ≤ |Fσ (z + s) − Fσ (z)| dz ds ≤ sup |Fσ0 (z + s) − Fσ0 (z)| dz || 0 |s|≤|| x

x

113

(2.124)

and the fact that the translations are continuous in L1 . Therefore the following lemma holds Lemma 2.49. Kx,σ and thus (Id + Kx,σ )−1 is a family of operators from L2y≥0 to L2y≥0 which depends continuously on the parameter x. Moreover the map (c, ∞) → L(L2y≥0 ), x 7→ Kx,σ is 0 differentiable and its derivative is the operator Kx,σ defined in (2.123). 0 Lemma 2.50. Let Fσ satisfy assumption (H), and gσ ∈ Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 be such that 2 kgσ kL2 L2 ≤ Kc kσkH 4 ∩L2 and S 4,N → L2x≥c L2y≥0 , σ 7→ gσ be real analytic. Then x≥c

y≥0

N

ζ,C

FR (x, y) :=

+∞ Z Fσ (x + y + z) gσ (x, z) dz 0

satisfies (P ). Proof. (P 1) For 6= 0 sufficiently small kFR (x + , ·) − FR (x, ·)kL2

y≥0

≤ kFσ (x + , ·) − Fσ (x, ·)kL1 kgσ (x + , ·)kL2

y≥0

+ kFσ kL1 kgσ (x + , ·) − gσ (x, ·)kL2

y≥0

which goes to 0 as → 0, proving that FR ∈ FR ∈ L2x≥c L2y≥0 and fulfills kFR kL2

x≥c

L2y≥0

0 Cx≥c L2y≥0 .

≤ kFσ kL1 kgσ kL2

x≥c

Furthermore, by Lemma 2.42 (A4), 2

L2y≥0

≤ Kc kσkH 4

2 ζ,C ∩LN

.

(2.125)

0 Now we show that FR ∈ Cx≥c,y≥0 . Let (xn )n≥1 ⊆ [c, ∞) and (yn )n≥1 ⊆ [0, ∞) be two sequences such that xn → x0 , yn → y0 . Then Fσ (xn + yn + ·)gσ (xn , ·) → Fσ (x0 + y0 + ·)gσ (x0 , ·) in L1z≥0 as n → ∞. Indeed

kFσ (xn + yn + ·)gσ (xn , ·) − Fσ (x0 + y0 + ·)gσ (x0 , ·)kL1

≤

z≥0

≤ kFσ (xn + yn + ·) − Fσ (x0 + y0 + ·)kL2

z≥0

+ kFσ (x0 + y0 + ·)kL2

z≥0

kgσ (xn , ·)kL2

y≥0

kgσ (xn , ·) − gσ (x0 , ·)kL2

,

y≥0

and the r.h.s. of the inequality above goes to 0 as (xn , yn ) → (x0 , y0 ), by the continuity of the 0 translations in L2 and the fact that gσ ∈ Cx≥c L2y≥0 . Thus it follows that FR (xn , yn ) → FR (x0 , y0 ) 0 as n → ∞, i.e., FR ∈ Cx≥c,y≥0 . We evaluate FR at y = 0, getting FR (x, 0) =

+∞ Z Fσ (x + z)gσ (x, z) dz . 0

By Lemma 2.42 (A2), FR (·, 0) ∈ kFR (·, 0)kL2

L2x≥c

x≥c

and fulfills

≤ kFσ kL2 kgσ kL2

x≥c

2

L2y≥0

≤ Kc kσkH 4

2 ζ,C ∩LN

.

(2.126)

(P 2) It follows from (2.125) and (2.126). (P 3) It follows by Lemma 2.42 (A2) and the fact that FR and FR (·, 0) are composition of real analytic maps. 114

We study now the solution of equation (2.119). Lemma 2.51. Assume that hσ satisfies (P ) and Fσ satisfies (H). Then equation (2.119) has a 0 unique solution gσ in Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 which satisfies (P ). Proof. We start to show that gσ exists and satisfies (P 1). Since hσ satisfies (P ) and Fσ satisfies (H), it follows that for any x ≥ c, gσ (x, ·) := (Id + Kx,σ )−1 [hσ (x, ·)] is the unique solution in L2y≥0 of the integral equation (2.119). Furthermore, by (2.122), kgσ (x, ·)kL2 ≤ Cσ khσ (x, ·)kL2 , which y≥0 y≥0 implies kgσ kL2 L2 ≤ Cσ khσ kL2 L2 . (2.127) x≥c

y≥0

0 Since hσ ∈ Cx≥c L2y≥0 , Lemma 2.49 implies 0 2 gσ ∈ Cx≥c Ly≥0 ∩ L2x≥c L2y≥0 . Now write

x≥c

that gσ ∈

y≥0

0 Cx≥c L2y≥0

as well. Thus we have proved that

+∞ Z gσ (x, y) = hσ (x, y) − Fσ (x + y + z)gσ (x, z) dz .

(2.128)

0

By Lemma 2.50 and the assumption that hσ satisfies (P ), it follows that the r.h.s. of formula (2.128) satisfies (P ).

The following lemma will be useful in the following: Lemma 2.52.

0 (i) Let Fσ satisfy (H), and gσ ∈ Cx≥c L2y≥0 ∩L2x≥c L2y≥0 be such that kgσ kL2

x≥c

2

Kc kσkH 4

2 ζ,C ∩LN

L2y≥0

≤

and S 4,N → L2x≥c L2y≥0 , σ 7→ gσ be real analytic. Denote +∞ Z Φσ (x, y) := Fσ0 (x + y + z)gσ (x, z) dz .

(2.129)

0 0 Then Φσ ∈ Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 , the map S 4,N → L2x≥c L2y≥0 , σ 7→ Φσ is real analytic and

kΦσ kL2

x≥c

2

L2y≥0

≤ Kc kσkH 4

2 ζ,C ∩LN

,

(2.130)

4 where Kc > 0 depends locally uniformly on σ ∈ Hζ,C ∩ L2N . 0 (ii) Let gσ as in item (i), and furthermore let gσ ∈ Cx≥c,y≥0 and gσ (·, 0) ∈ L2x≥c . Assume furthermore that ∂y gσ satisfies the same assumptions as gσ in item (i). Then Φσ , defined in (2.129), satisfies (P ). 1 (iii) Assume that Fσ satisfies (H) and that the map S 4,N → Hx≥c , σ 7→ bσ is real analytic with 2 kbσ kH 1 ≤ Kc kσkH 4 ∩L2 . Then the function x≥c

ζ,C

N

φσ (x, y) := Fσ (x + y)bσ (x) satisfies (P ).

115

Proof. (i) Clearly kΦσ (x, ·)kL2

y≥0

Φσ ∈ L2x≥c L2y≥0 with kΦσ kL2

x≥c

that Φσ ∈

0 Cx≥c L2y≥0 .

≤ kFσ0 kL1 kgσ (x, ·)kL2

y≥0

L2y≥0

≤ kFσ0 kL2 kgσ kL2 4

x≥c

, and since gσ ∈ L2x≥c L2y≥0 it follows that

L2y≥0 ,

which implies (2.129). We show now

For 6= 0 one has

kΦσ (x + , ·) − Φσ (x, ·)kL2

y≥0

≤ kFσ0 (· + ) − Fσ0 kL1 kgσ (x, ·)kL2

y≥0

+ kFσ0 kL1 kgσ (x + , ·) − gσ (x, ·)kL2

.

y≥0

0 The continuity of the translation in L1 and the assumption gσ ∈ Cx≥c L2y≥0 imply that kΦσ (x + , ·) − Φσ (x, ·)kL2

y≥0

0 Cx≥c L2y≥0 .

0 as → 0, thus Φσ ∈ The real analyticity of σ 7→ Φσ follows from Lemma 2.42 (A4) and the fact that Φσ is composition of real analytic maps. (ii) Fix x ≥ c and use integration by parts to write +∞ Z Φσ (x, y) = −Fσ (x + y)gσ (x, 0) − Fσ (x + y + z)∂z gσ (x, z) dz ,

(2.131)

0 1 where we used that since Fσ ∈ H 1 [g(x, ·) ∈ Hy≥0 ], limx→∞ Fσ (x) = 0 [limy→∞ gσ (x, y) = 0]. By 0 the assumption and the proof of Lemma 2.50 (P 1), Φσ ∈ Cx≥c,y≥0 . We evaluate (2.131) at y = 0 to get the formula +∞ Z Φσ (x, 0) = −Fσ (x)gσ (x, 0) − Fσ (x + z)∂z gσ (x, z) dz . 0

Together with Lemma 2.42 (A2) we have the estimate kΦσ (·, 0)kL2 ≤ kFσ kH 1 kgσ (·, 0)kL2 + k∂y gσ kL2 x≥c

L2 x≥c y≥0

x≥c

2

≤ Kc kσkH 4

2 ζ,C ∩LN

.

(2.132)

Estimate (2.132) together with estimate (2.130) imply that Φσ satisfies (P 2). Finally σ 7→ Φσ (·, 0) is real analytic, being a composition of real analytic maps. (iii) We skip an easy proof. If the function hσ is more regular one deduces better regularity properties of the corresponding solution of (2.119). Lemma 2.53. Consider the integral equation (2.119) and assume that Fσ satisfies (H). Assume that hσ , ∂x hσ , ∂y hσ satisfy (P ). Then gσ solution of (2.119) satisfies (P ). Its derivatives ∂x gσ and ∂y gσ satisfy (P ) and solve the equations 0 (Id + Kx,σ ) [∂x gσ ] = ∂x hσ − Kx,σ [gσ ] ,

(2.133)

0 Kx,σ [gσ ]

(2.134)

∂y gσ = ∂y hσ −

.

Proof. By Lemma 2.51, gσ satisfies (P ). ∂y gσ satisfies (P ). For 6= 0 sufficiently small, we have in L2y≥0 gσ (x, y + ) − gσ (x, y) = Ψσ (x, y) 116

→

where Ψσ (x, y) :=

hσ (x, y + ) − hσ (x, y) −

+∞ Z

Fσ (x + y + + z) − Fσ (x + y + z) gσ (x, z) dz . (2.135)

0

Define Ψ0σ (x, y)

+∞ Z Fσ0 (x + y + z)gσ (x, z) dz . := ∂y hσ (x, y) − 0

0 Since ∂y hσ and gσ satisfy (P ), by Lemma 2.52 (i) it follows that Ψ0σ ∈ Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 , the

0 2 4,N 2 2 0 map S → Lx≥c Ly≥0 , σ → 7 Ψσ is real analytic and Ψσ L2 L2 ≤ Kc kσkH 4 ∩L2 . Furtherx≥c

y≥0

ζ,C

N

more one verifies that gσ (x, · + ) − gσ (x, ·) = lim Ψσ (x, ·) = Ψ0σ (x, ·) →0 →0

∂y gσ (x, ·) = lim

in L2y≥0 .

Thus ∂y gσ fulfills +∞ Z ∂y gσ (x, y) = ∂y hσ (x, y) − Fσ0 (x + y + z)gσ (x, z)dz ,

(2.136)

0

i.e., ∂y gσ satisfies equation (2.134). Since ∂y gσ = Ψ0σ , gσ satisfies the assumptions of Lemma 2.52 (ii). Since ∂y hσ satisfies (P ) as well, it follows that ∂y gσ satisfies (P ). ∂x gσ satisfies (P ). For 6= 0 small enough we have in L2y≥0 gσ (x + , ·) − gσ (x, ·) = Φσ (x, ·) (Id + Kx+,σ ) where Φσ (x, y)

hσ (x + , y) − hσ (x, y) − :=

+∞ Z

Fσ (x + y + + z) − Fσ (x + y + z) gσ (x, z) dz .

0

Define Φ0σ (x, y)

+∞ Z := ∂x hσ (x, y) − Fσ0 (x + y + z)gσ (x, z) dz . 0

Proceeding as above, one proves that Φ0σ satisfies (P ), and lim Φσ (x, ·) = Φ0σ (x, ·)

→0

in L2y≥0 .

Together with Lemma 2.49 we get for x > c in L2y≥0 ∂x gσ (x, ·) = lim

→0

gσ (x + , ·) − gσ (x, ·) −1 −1 = lim (Id + Kx+,σ ) Φσ (x, ·) = (Id + Kx,σ ) Φ0σ (x, ·) . →0 (2.137) 117

In particular (Id + Kσ )(∂x gσ (x, ·)) = Φ0σ (x, ·). Since Φ0σ satisfies (P ), by Lemma 2.51, ∂x gσ satisfies (P ). Formula (2.137) implies that +∞ +∞ Z Z Fσ0 (x + y + z)gσ (x, z) dz , (2.138) Fσ (x + y + z)∂x gσ (x, z) dz = ∂x hσ (x, y) − ∂x gσ (x, y) + 0

0

namely ∂x gσ satisfies equation (2.133).

D D.1

Proof from Section 4 ± and f±,σ . Properties of Kx,σ

± and f±,σ , defined in (2.94) and (2.96), which will We begin with proving some properties of Kx,σ be needed later. ± Properties of Id + Kx,σ . In order to solve the integral equations (2.93) we need the operator − + to be invertible on L2y≤0 ). The following to be invertible on L2y≥0 (respectively Id + Kx,σ Id + Kx,σ result is well known:

Lemma 2.54 ([DT79, CK87a]). Let σ ∈ S 4,0 and fix c ∈ R. Then the following holds: + (i) For every x ≥ c, Kx,σ : L2y≥ 0 → L2y≥ 0 is a bounded linear operator; moreover

+

2 sup Kx,σ L(L

y≥0

x≥c

)

< 1,

and

+

Kx,σ 2 L(L

y≥0

)

+∞ Z ≤ |F+,σ (ξ)| dξ → 0

if

x → +∞.

x

(2.139) + (ii) The map Kσ+ : L2x≥c L2y≥0 → L2x≥c L2y≥0 , f 7→ Kσ+ [f ], where Kσ+ [f ](x, y) := Kx,σ [f ](y), is linear + 2 2 and bounded. Moreover the operators Id ± Kσ are invertible on Lx≥c Ly≥0 and there exists a constant Kc > 0, which depends locally uniformly on σ ∈ S 4,0 , such that

−1

≤ Kc . (2.140)

Id ± Kσ+

2 2 L(Lx≥c Ly≥0 )

−1

(iii) σ 7→ (Id ± Kσ+ )

are real analytic as maps from S 4,0 to L(L2x≥c L2y≥0 ).

− Analogous results hold also for Kx,σ replacing L2x≥c L2y≥0 by L2x≤c L2y≤0 .

Properties of f±,σ . First note that f±,σ , defined by (2.96), are well defined. Indeed for any σ ∈ S 4,0 , Proposition 2.33 implies that F±,σ ∈ H 1 ∩ L23 ⊂ L2 . Hence for any x ≥ c, y ≥ 0 the map given by z 7→ F+,σ (x + y + z)F+,σ (x + z) is in L1z≥0 . Similarly, for any x ≥ c, y ≥ 0, the map given by z 7→ F−,σ (x + y + z)F−,σ (x + z) is in L1z≤0 . 118

In the following we will use repeatedly the Hardy inequality [HLP88]

+∞

Z

m

hxi ≤ Kc hxim+1 g L2 , ∀m ≥ 0 . g(z)dz

x≥c

2

x

(2.141)

Lx≥c

The inequality is well known, but for sake of completeness we give a proof of it in Lemma 2.42 (A3). We analyze now the maps σ 7→ f±,σ . Since the analysis of f+,σ and the one of f−,σ are similar, we will consider f+,σ only. To shorten the notation we will suppress the subscript ” + ” in what follows. Lemma 2.55. Fix N ∈ Z≥0 and let σ ∈ S 4,N . Let fσ ≡ f+,σ be given as in (2.96). Then for every j1 , j2 ∈ Z≥0 with 0 ≤ j1 + j2 ≤ N + 1, the function ∂xj1 ∂yj2 fσ satisfies (P ). Proof. We prove at the same time (P 1), (P 2) and (P 3) for any j1 , j2 ≥ 0 with j1 + j2 = n for any 0 ≤ n ≤ N + 1. Case n = 0. Then j1 = j2 = 0. By Proposition 2.33, for any N ∈ Z≥0 one has Fσ ≡ F+,σ ∈ H 1 ∩L23 . 0 (P 1) We show that fσ ∈ Cx≥c L2y≥0 . For any x ≥ c fixed one has kfσ (x, ·)kL2

y≥0

which shows that fσ (x, ·) ∈

L2y≥0 .

≤ kFσ kL1 kFσ (x + ·)kL2

y≥0

For 6= 0 sufficiently small one has

kfσ (x + , ·) − fσ (x, ·)kL2

x≥c

≤ kFσ kL1 kFσ (x + + ·) − Fσ (x + ·)kL2

y≥0

+ kFσ ( + ·) − Fσ kL1 kFσ (x + ·)kL2

y≥0

p

which goes to 0 as → 0, due to the continuity of the translations in L -space, 1 ≤ p < ∞. 0 Thus fσ ∈ Cx≥c L2y≥0 . We show now that fσ ∈ L2x≥c L2y≥0 . Introduce hσ (x, y) := Fσ (x + y). Then hσ ∈ L2x≥c L2y≥0 , since for some C, C 0 > 0 khσ kL2

x≥c

≤ C kFσ kL2

L2y≥0

1/2,x≥c

≤ C 0 kσkH 4

(2.142)

ζ,C

where for the first [second] inequality we used Lemma 2.42 (A0) [Proposition 2.33 (i)]. By Lemma 2.42(A4) and using once more Proposition 2.33 (i), one gets kfσ kL2

x≥c

L2y≥0

≤ C 00 kFσ kL1

x≥c

khσ kL2

x≥c

L2y≥0

≤ C 000 kFσ kL2 khσ kL2 1

x≥c

2

L2y≥0

≤ C 0000 kσkH 4

,

ζ,C

(2.143) for some C 00 , C 000 , C 0000 > 0. Thus fσ ∈ L2x≥c L2y≥0 . 0 To show that fσ ∈ Cx≥c,y≥0 proceed as in Lemma 2.50. +∞ R Finally we show that fσ (·, 0) ∈ L2x≥c . Evaluate (2.96) at y = 0 to get fσ (x, 0) = Fσ2 (z) dz. x R +∞ Using the Hardy inequality (2.141), Fσ (x) = − x Fσ0 (s) ds and Proposition 2.33 one obtains

kfσ (·, 0)kL2 ≤ hxiFσ2 L2 ≤ khxiFσ kL∞ kFσ kL2 ≤ Kc khxiFσ0 kL1 kFσ kL2 x≥c x≥c x≥c x≥c x≥c x≥c

2 0 2 0 00

≤ Kc hxi Fσ L2 kFσ kL2 ≤ Kc kσkH 4 ∩L2 , (2.144) x≥c

x≥c

for some constants Kc , Kc0 , Kc00 > 0. Thus fσ (·, 0) ∈ L2x≥c . 119

ζ,C

N

,

(P 2) It follows from (2.143) and (2.144). (P 3) By Proposition 2.33 (i), S 4,0 → HC1 ∩ L23 , σ 7→ Fσ is real analytic and by Lemma 2.42 (A0) so is S 4,0 → L2x≥c L2y≥0 , σ 7→ hσ . By Lemma 2.42 (A4) it follows that S 4,0 → L2x≥c L2y≥0 , σ 7→ fσ is real analytic. Since the map σ 7→ fσ (·, 0) is a composition of real analytic maps, it is real analytic as a map from S 4,N to L2x≥c . Case n ≥ 1. By Proposition 2.33, Fσ ∈ H N +1 and kFσ kH N +1 ≤ C 0 kσkH 4

2 ζ,C ∩LN

. By Sobolev

1 2.

Moreover since limx→+∞ Fσ (x) = 0, embedding theorem, it follows that Fσ ∈ C N,γ (R, R), γ < one has +∞ Z Fσ (y + z)Fσ (z) dz = −Fσ (x + y)Fσ (x) . (2.145) ∂x fσ (x, y) = ∂x x

Consider first the case j1 ≥ 1. Then j2 ≤ N . By (2.145) it follows that ∂xj1 ∂yj2 fσ (x, y)

=−

jX 1 −1

j1 − 1 Fσ(j2 +l) (x + y)Fσ(j1 −1−l) (x) , l

l=0 (l)

where Fσ bσ = (P ).

(2.146)

≡ ∂xl Fσ . Thus ∂xj1 ∂yj2 fσ is a linear combination of terms of the form (2.148), with

(j −1−l) Fσ 1

satisfying the assumption of Lemma 2.56 (i), thus ∂xj1 ∂yj2 fσ , with j1 ≥ 1, satisfies

Consider now the case j1 = 0. Then 1 ≤ j2 ≤ n ≤ N + 1. Since ∂y Fσ (x + y + z) = ∂z Fσ (x + y + z) = Fσ0 (x + y + z), by integration by parts one obtains ∂yj2 fσ (x, y)

=

−Fσ(j2 −1) (x

+∞ Z + y)Fσ (x) − Fσ(j2 −1) (x + y + z)Fσ0 (x + z)dz .

(2.147)

0

Then, by Lemma 2.56 (i) and (ii), (P ) as well.

∂yj2 fσ

is the sum of two terms which satisfy (P ), thus it satisfies

Lemma 2.56. Fix c ∈ R, N ∈ Z≥0 and let σ ∈ S 4,N . Let Fσ be given as in (2.76). Then the following holds true: 1 (i) Let σ 7→ bσ be real analytic as a map from S 4,N to Hx≥c , satisfying kbσ kH 1

x≥c

4 Hζ,C

≤ Kc kσkH 4

2 ζ,C ∩LN

L2N .

Then for every integer k with where Kc > 0 depends locally uniformly on σ ∈ ∩ 0 ≤ k ≤ N , the function HR (x, y) := Fσ(k) (x + y) bσ (x) (2.148) satisfies (P ). (ii) For every integer 0 ≤ k ≤ N , the function +∞ Z GR (x, y) = Fσ(k) (x + y + z)Fσ0 (x + z)dz 0

satisfies (P ). 120

(2.149)

,

(iii) Let N ≥ 1 and let Gσ be a function satisfying (P ). Then the function +∞ Z Fσ0 (x + y + z)Gσ (x, z) dz FR (x, y) :=

(2.150)

0

satisfies (P ). Proof. (i) HR satisfies (P 1). Clearly HR (x, ·) ∈ L2y≥0 and by the continuity of the translations in 0 L2 one verifies that kHR (x + , ·) − HR (x, ·)kL2 → 0 as → 0, thus proving that HR ∈ Cx≥c L2y≥0 . y≥0

We show now that HR ∈ L2x≥c L2y≥0 . By Lemma 2.42 (A1), Proposition 2.33 and the assumption on bσ , one has that kHR kL2

x≥c

L2y≥0

≤ C kFσ kH N +1 kbσ kL2

x≥c

2

≤ Kc kσkH 4

2 ζ,C ∩LN

,

(2.151)

4 where Kc > 0 can be chosen locally uniformly for σ ∈ Hζ,C ∩ L2N . (k)

0 . For 0 ≤ k ≤ N , Fσ ∈ C 0 (R, R) by the Sobolev embedding theorem. Thus HR ∈ Cx≥c,y≥0 2 Finally we show that HR (·, 0) ∈ Lx≥c . We evaluate the r.h.s. of formula (2.148) at y = 0, getting

HR (x, 0) = Fσ(k) (x)bσ (x) . 4 ∩ L2N , such It follows that there exists C > 0 and Kc > 0, depending locally uniformly on σ ∈ Hζ,C that 2 kHR (·, 0)kL2 ≤ C kFσ kH N +1 kbσ kH 1 ≤ Kc kσkH 4 ∩L2 , (2.152) ζ,C

x≥c

x≥c

N

(k) Fσ

1 and bσ are in Hx≥c . where we used that both HR satisfies (P 2). It follows from (2.151) and (2.152). HR satisfies (P 3). The real analyticity property follows from Lemma 2.42 and Proposition 2.33, since for every 0 ≤ k ≤ N , HR is product of real analytic maps.

(ii) GR satisfies (P 1). We show that GR ∈ L2x≥c L2y≥0 . By Lemma 2.42 (A5) and Proposition 2.33 it follows that kGR kL2

x≥c

2

L2y≥0

≤ kFσ kH N +1 kFσ0 kL1 ≤ Kc kσkH 4

2 ζ,C ∩LN

,

(2.153)

0 4 L2y≥0 . where Kc > 0 depends locally uniformly on σ ∈ Hζ,C ∩L2N . One verifies easily that GR ∈ Cx≥c 0 In order to prove that GR ∈ Cx≥c,y≥0 , proceed as in Lemma 2.50. Now we show that GR (·, 0) ∈ L2x≥c . We evaluate formula (2.149) at y = 0 getting that

Z GR (x, 0) =

∞

Fσ(k) (x + z)Fσ0 (x + z)dz .

0

Let

h0σ (x, z)

:=

Fσ0 (x

+ z). By Lemma 2.42 (A0) and Proposition 2.33 one has

kh0σ kL2 L2 ≤ hxi1/2 Fσ0 2 ≤ Kc kσkH 4 ∩L2 , N ζ,C x≥c

y≥0

Lx≥c

121

4 where Kc > 0 can be chosen locally uniformly for σ ∈ Hζ,C ∩ L2N . Thus by Lemma 2.42 (A2) one gets

2 (2.154) kGR (·, 0)kL2 ≤ Kc Fσ(k) 2 kh0σ kL2 L2 ≤ Kc kσkH 4 ∩L2 , N ζ,C x≥c

x≥c

Lx≥c

y≥0

4 where Kc > 0 can be chosen locally uniformly for σ ∈ Hζ,C ∩ L2N . GR satisfies (P 2). It follows from (2.153) and (2.154). GR satisfies (P 3). The real analyticity property follows from Lemma 2.42 and Proposition 2.33, since for every 0 ≤ k ≤ N , GR is composition of real analytic maps. 0 (iii) FR satisfies (P 1). By Lemma 2.52 (i), FR ∈ Cx≥c L2y≥0 ∩ L2x≥c L2y≥0 and

kFR kL2

x≥c

L2y≥0

2

≤ kFσ0 kL2 kGσ kL2 4

x≥c

L2y≥0

≤ Kc kσkH 4

2 ζ,C ∩LN

.

0 Proceeding as in the proof of Lemma 2.55 (P 1) one shows that FR ∈ Cx≥c,y≥0 . Since Fσ0 ∈ H N , 0 N ≥ 1, Fσ is a continuous function. Thus we can evaluate FR at y = 0, obtaining FR (x, 0) = +∞ R Fσ0 (x + z)Gσ (x, z) dz. By Lemma 2.42 (A2) we have that 0

kFR (·, 0)kL2

x≥c

≤ kFσ0 kL2

x≥c

kGσ kL2

x≥c

2

L2y≥0

≤ Kc kσkH 4

2 ζ,C ∩LN

.

The proof that FR satisfies (P 2) and (P 3) follows as in the previous items. We omit the details. Lemma 2.57. Let N ≥ 1 be fixed. For every j1 , j2 ≥ 0 with 1 ≤ j1 + j2 ≤ N , the function fσj1 ,j2 defined in (2.99) and its derivatives ∂y fσj1 ,j2 , ∂x fσj1 ,j2 satisfy (P ). Proof. First note that by Lemma 2.55 the terms ∂xj1 ∂yj2 fσ and its derivatives ∂xj1 +1 ∂yj2 fσ , ∂xj1 ∂yj2 +1 fσ satisfy (P ). It thus remains to show that FkR1 ,k2 (x, y)

+∞ Z := ∂xk1 Fσ (x+y +z) ∂xk2 Bσ (x, z) dz ,

k1 ≥ 1, k2 ≥ 0,

k1 +k2 = n ≤ N (2.155)

0

and its derivatives ∂y FkR1 ,k2 , ∂x FkR1 ,k2 satisfy (P ). Remark that, by the induction assumption in the proof of Lemma 2.38, for every integers k1 , k2 ≥ 0 with k1 + k2 ≤ n, ∂xk1 ∂yk2 Bσ satisfies (P ). FkR1 ,k2 satisfies (P ). If k1 = 1, it follows by Lemma 2.56 (iii). Let k1 > 1. By integration by parts k1 − 1 times we obtain FkR1 ,k2 (x, y) =

kX 1 −1

(−1)l ∂xk1 −l Fσ (x + y)(∂xk2 ∂zl−1 Bσ )(x, 0)

l=1

+ (−1)k1 −1

+∞ Z Fσ0 (x + y + z) ∂xk2 ∂zk1 −1 Bσ (x, z) dz ,

(2.156)

0 (k1 −l)

where we used that for 1 ≤ l ≤ k1 − 1 one has Fσ (k1 −l)

limx→∞ Fσ

1 ∈ H 1 [(∂xk2 ∂yl−1 Bσ )(x, ·) ∈ Hy≥0 ], thus

(x) = 0 [limy→∞ ∂xk2 ∂yl−1 Bσ )(x, y) = 0]. Consider the r.h.s. of (2.156). It is a 122

linear combinations of terms of the form (2.148) and (2.150). By the induction assumption, these k1 ,k2 terms satisfy the hypothesis of Lemma 2.56 (i) and (iii). It follows that FR satisfies (P ), and in 4 particular there exists a constant Kc > 0, depending locally uniformly on σ ∈ Hζ,C ∩ L2N , such that

k1 ,k2

FR

L2x≥c L2y≥0

+ FkR1 ,k2 (·, 0)

2

L2x≥c

≤ Kc kσkH 4

2 ζ,C ∩LN

.

(2.157)

∂y FkR1 ,k2 satisfies (P ). For 6= 0 sufficiently small, by integration by parts k1 -times we obtain k

1 FkR1 ,k2 (x, y + ) − FkR1 ,k2 (x, y) X ∂ k1 −l Fσ (x + y + ) − ∂xk1 −l Fσ (x + y) k2 l−1 = (−1)l x (∂x ∂z Bσ )(x, 0)

l=1

+ (−1)

k1

+∞ Z

Fσ (x + y + + z) − Fσ (x + y + z) k2 k1 ∂x ∂z Bσ (x, z) dz ,

0 (k1 −l)

where once again we used that for 1 ≤ l ≤ k1 one has Fσ (k1 −l)

thus limx→∞ Fσ

1 ∈ H 1 [(∂xk2 ∂yl−1 Bσ )(x, ·) ∈ Hy≥0 ],

(x) = 0 [limy→∞ ∂xk2 ∂yl−1 Bσ )(x, y) = 0]. Define also

∂y FkR1 ,k2 (x, y) :=

k1 X (−1)l ∂xk1 −l+1 Fσ (x + y) (∂xk2 ∂zl−1 Bσ )(x, 0) l=1

+ (−1)k1

+∞ Z Fσ0 (x + y + z) ∂xk2 ∂zk1 Bσ (x, z) dz .

(2.158)

0

Consider the r.h.s. of equation (2.158). It is a linear combinations of terms of the form (2.148) and (2.150). By the induction assumption, these terms satisfy the hypothesis of Lemma 2.56 (i) and (iii). It follows that ∂y FkR1 ,k2 satisfies (P ) and one has

2 k ,k k1 ,k2 + F (·, 0) (2.159)

∂y FR1 2 2

∂

2 ≤ Kc0 kσkHζ,C 4 ∩L2 y R 2 N Lx≥c Ly≥0

Lx≥c

4 ∩ L2N . Furthermore one verifies for some constant Kc0 > 0, depending locally uniformly on σ ∈ Hζ,C that k1 ,k2 Fk1 ,k2 (x, · + ) − FR (x, ·) k1 ,k2 lim R (x, ·) in L2y≥0 . = ∂y FR →0

∂x FkR1 ,k2 satisfies (P ). The proof is similar to the previous case, and the details are omitted. This conclude the proof of the inductive step.

E

Hilbert transform

Define H : L2 (R, C) → L2 (R, C) as the Fourier multiplier operator \ (H(v))(ξ) = −i sign(ξ) vˆ(ξ) .

123

Thus H is an isometry on L2 (R, C). It is easy to see that H|H N : HCN → HCN is an isometry for C any N ≥ 1 – cf. [Duo01]. In case v ∈ C 1 (R, C) with kv 0 kL∞ , kxv(x)kL∞ < ∞, one has Z 1 v(k 0 ) 0 H(v)(k) = − lim+ dk π →0 |k0 −k|≥ k 0 − k and obtains the estimate |H(v)(k)| ≤ C(kv 0 k∞ +kxv(x)k∞ ), where C > 0 is a constant independent of v and k. Let g ∈ C 1 (R, R) with kg 0 kL∞ , kxg(x)kL∞ < ∞. Then define for z ∈ C+ := {z ∈ C : Im(z) > 0} the function Z ∞ g(s) 1 ds . f (z) := πi −∞ s − z Decompose

1 s−z

into real and imaginary part b 1 1 s−a = = +i s−z s − a − ib (s − a)2 + b2 (s − a)2 + b2

to get the formulas for the real and imaginary part of f (z) Z 1 ∞ b Re f (z) = g(s)ds , π −∞ (s − a)2 + b2 Z −1 ∞ s−a Im f (z) = g(s)ds . π −∞ (s − a)2 + b2

(2.160) (2.161)

The following Lemma is well known and can be found in [Duo01]. Lemma 2.58. The function f is analytic and admits a continuous extension to the real line. Furthermore it has the following properties for any a ∈ R: (i) limb→0+ Im f (a + bi) = H(g)(a). (ii) limb→0+ Re f (a + bi) = g(a). (iii) There exists C > 0 such that |f (z)| ≤

C 1+|z| ,

∀ z ∈ {z : Im z ≥ 0}.

(iv) Let f˜(z) be a continuous function on Im z ≥ 0 which is analytic on Im z > 0 and satisfies 1 Re f˜|R = g and |f˜(z)| = O( |z| ) as |z| → ∞, then f˜ = f . The next lemma follows from the commutator estimates due to Calderón [Cal65]: Lemma 2.59 ([Cal65]). Let b : R → R have first-order derivative in L∞ . For any p ∈ (1, ∞) there exists C > 0, such that k[H, b] ∂x gkLp ≤ C kgkLp . We apply this lemma to prove the following result: M M Lemma 2.60. Let M ∈ Z≥1 be fixed. Then H : Hζ,C → Hζ,C is a bounded linear operator.

124

M Proof. Let f ∈ Hζ,C . As the Hilbert transform commutes with the derivatives, we have that M −1 H(f ) ∈ HC . Next we show that if ζ∂kM f ∈ L2 , then ζ∂kM H(f ) = ζH(∂kM f ) ∈ L2 . By Lemma 2.59 with p = 2, g = ∂kM −1 f and b = ζ, we have that

ζH(∂kM f ) 2 ≤ H(ζ∂kM f ) 2 + [H, ζ] ∂kM f 2 ≤ kf k M + C ∂ M −1 f 2 < ∞ . H k L L L L ζ,C

125

Bibliography [AC91] M. J. Ablowitz and P. A. Clarkson. Solitons, nonlinear evolution equations and inverse scattering, volume 149 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1991. [Ahl78] L. V. Ahlfors. Complex analysis. McGraw-Hill Book Co., New York, third edition, 1978. An introduction to the theory of analytic functions of one complex variable, International Series in Pure and Applied Mathematics. [AKNS74] M. Ablowitz, D. Kaup, A. Newell, and H. Segur. The inverse scattering transformFourier analysis for nonlinear problems. Studies in Appl. Math., 53(4):249–315, 1974. [BCP13] G. Benettin, H. Christodoulidi, and A. Ponno. The Fermi-Pasta-Ulam problem and its underlying integrable dynamics. J. Stat. Phys., 152(2):195–212, 2013. [BG06] D. Bambusi and B. Grébert. Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J., 135(3):507–567, 2006. [BGG04] L. Berchialla, L. Galgani, and A. Giorgilli. Localization of energy in FPU chains. Discrete Contin. Dyn. Syst., 11(4):855–866, 2004. [BGGK93] D. Bättig, B. Grébert, J.-C. Guillot, and T. Kappeler. Fibration of the phase space of the periodic Toda lattice. J. Math. Pures Appl. (9), 72(6):553–565, 1993. [BGP04] L. Berchialla, A. Giorgilli, and S. Paleari. Exponentially long times to equipartition in the thermodynamic limit. Physics Letters A, 321(3):167 – 172, 2004. [BGPU03] A. Bloch, F. Golse, T. Paul, and A. Uribe. Dispersionless Toda and Toeplitz operators. Duke Math. J., 117(1):157–196, 2003. [BIT11] A. Babin, A. Ilyin, and E. Titi. On the regularization mechanism for the periodic Korteweg-de Vries equation. Comm. Pure Appl. Math., 64(5):591–648, 2011. [BKP09] D. Bambusi, T. Kappeler, and T. Paul. De Toda à KdV. C. R. Math. Acad. Sci. Paris, 347(17-18):1025–1030, 2009. [BKP13a] D. Bambusi, T. Kappeler, and T. Paul. Dynamics of periodic Toda chains with a large number of particles. ArXiv e-prints, arXiv:1309.5441 [math.AP], September 2013. [BKP13b] D. Bambusi, T. Kappeler, and T. Paul. From Toda to KdV. arXiv:1309.5324 [math.AP], September 2013. 126

ArXiv e-prints,

[BM14] D. Bambusi and A. Maspero. Birkhoff coordinates for the Toda Lattice in the limit of infinitely many particles with an application to FPU. ArXiv e-prints, July 2014. [BP06] D. Bambusi and A. Ponno. On metastability in FPU. Comm. Math. Phys., 264(2):539– 561, 2006. [BP11] G. Benettin and A. Ponno. Time-scales to equipartition in the Fermi-Pasta-Ulam problem: finite-size effects and thermodynamic limit. J. Stat. Phys., 144(4):793–812, 2011. [Bre11] H. Brezis. Functional analysis, Sobolev spaces and partial differential equations. Universitext. Springer, New York, 2011. [BS75] J. L. Bona and R. Smith. The initial-value problem for the Korteweg-de Vries equation. Philos. Trans. Roy. Soc. London Ser. A, 278(1287):555–601, 1975. [Cal65] A. Calderón. Commutators of singular integral operators. Proc. Nat. Acad. Sci. U.S.A., 53:1092–1099, 1965. [Car07] A. Carati. An averaging theorem for Hamiltonian dynamical systems in the thermodynamic limit. J. Stat. Phys., 128(4):1057–1077, 2007. [CK87a] A. Cohen and T. Kappeler. The asymptotic behavior of solutions of the Korteweg-de Vries equation evolving from very irregular data. Ann. Physics, 178(1):144–185, 1987. [CK87b] A. Cohen and T. Kappeler. Solutions to the Korteweg-de Vries equation with initial profile in L11 (R) ∩ L1N (R+ ). SIAM J. Math. Anal., 18(4):991–1025, 1987. [CM12] A. Carati and A. Maiocchi. Exponentially long stability times for a nonlinear lattice in the thermodynamic limit. Comm. Math. Phys., 314(1):129–161, 2012. [DT79] P. Deift and E. Trubowitz. Inverse scattering on the line. Comm. Pure Appl. Math., 32(2):121–251, 1979. [Duo01] J. Duoandikoetxea. Fourier analysis, volume 29 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. [Eli90] L. H. Eliasson. Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case. Comment. Math. Helv., 65(1):4–35, 1990. [ET13a] M. Erdoğan and N. Tzirakis. Global smoothing for the periodic KdV evolution. Int. Math. Res. Not. IMRN, (20):4589–4614, 2013. [ET13b] M. Erdoğan and N. Tzirakis. Long time dynamics for forced and weakly damped KdV on the torus. Commun. Pure Appl. Anal., 12(6):2669–2684, 2013. [Fad64] L. D. Faddeev. Properties of the S-matrix of the one-dimensional Schrödinger equation. Trudy Mat. Inst. Steklov., 73:314–336, 1964. [FFM82] W. E. Ferguson, Jr., H. Flaschka, and D. W. McLaughlin. Nonlinear normal modes for the Toda chain. J. Comput. Phys., 45(2):157–209, 1982. 127

[FHMP09] C. Frayer, R. O. Hryniv, Ya. V. Mykytyuk, and P. A. Perry. Inverse scattering for Schrödinger operators with Miura potentials. I. Unique Riccati representatives and ZS-AKNS systems. Inverse Problems, 25(11):115007, 25, 2009. [Fla74] H. Flaschka. The Toda lattice. I. Existence of integrals. Phys. Rev. B (3), 9:1924–1925, 1974. [FM76] H. Flaschka and D. W. McLaughlin. Canonically conjugate variables for the Kortewegde Vries equation and the Toda lattice with periodic boundary conditions. Progr. Theoret. Phys., 55(2):438–456, 1976. [FPU65] E. Fermi, J. Pasta, and S. Ulam. Studies of non linear problems. In Enrico Fermi Collected Papers, vol. II, pages 977–988. University of Chicago Press/Accademia Nazionale dei Lincei, Chicago/Roma, 1965. [FT87] L. Faddeev and L. Takhtajan. Hamiltonian methods in the theory of solitons. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987. Translated from the Russian by A. G. Reyman [A. G. Re˘ıman]. [FY01] G. Freiling and V. Yurko. Inverse Sturm-Liouville problems and their applications. Nova Science Publishers, Inc., Huntington, NY, 2001. [GGKM67] C. Gardner, J. Greene, M. Kruskal, and R. Miura. Method for Solving the KortewegdeVries Equation. Physical Review Letters, 19:1095–1097, November 1967. [GGKM74] C. Gardner, J. Greene, M. Kruskal, and R. Miura. Korteweg-deVries equation and generalization. VI. Methods for exact solution. Comm. Pure Appl. Math., 27:97–133, 1974. [GK14] B. Grebert and T. Kappeler. The Defocusing NLS Equation and Its Normal Form. EMS Series of Lectures in Mathematics. European Mathematical Society Publishing House, 2014. [GPP12] A. Giorgilli, S. Paleari, and T. Penati. Extensive adiabatic invariants for nonlinear chains. J. Stat. Phys., 148(6):1106–1134, 2012. [Hén74] M. Hénon. Integrals of the Toda lattice. Phys. Rev. B (3), 9:1921–1923, 1974. [HK08a] A. Henrici and T. Kappeler. Birkhoff normal form for the periodic Toda lattice. In Integrable systems and random matrices, volume 458 of Contemp. Math., pages 11–29. Amer. Math. Soc., Providence, RI, 2008. [HK08b] A. Henrici and T. Kappeler. Global action-angle variables for the periodic Toda lattice. Int. Math. Res. Not. IMRN, (11):Art. ID rnn031, 52, 2008. [HK08c] A. Henrici and T. Kappeler. Global Birkhoff coordinates for the periodic Toda lattice. Nonlinearity, 21(12):2731–2758, 2008. [HL12] E. Hairer and C. Lubich. On the energy distribution in Fermi-Pasta-Ulam lattices. Arch. Ration. Mech. Anal., 205(3):993–1029, 2012.

128

[HLP88] G. H. Hardy, J. E. Littlewood, and G. Pólya. Inequalities. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1988. Reprint of the 1952 edition. [HMP11] R. O. Hryniv, Y. V. Mykytyuk, and P. A. Perry. Sobolev mapping properties of the scattering transform for the Schrödinger equation. In Spectral theory and geometric analysis, volume 535 of Contemp. Math., pages 79–93. Amer. Math. Soc., Providence, RI, 2011. [Kap88] T. Kappeler. Inverse scattering for scattering data with poor regularity or slow decay. J. Integral Equations Appl., 1(1):123–145, 1988. [Kap91] T. Kappeler. Fibration of the phase space for the Korteweg-de Vries equation. Ann. Inst. Fourier (Grenoble), 41(3):539–575, 1991. [Kat66] T. Kato. Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, 1966. [Kat79] T. Kato. On the Korteweg-de Vries equation. Manuscripta Math., 28(1-3):89–99, 1979. [KLTZ09] T. Kappeler, P. Lohrmann, P. Topalov, and N. T. Zung. Birkhoff coordinates for the focusing NLS equation. Comm. Math. Phys., 285(3):1087–1107, 2009. [KP03] T. Kappeler and J. Pöschel. KdV & KAM, volume 45 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer-Verlag, Berlin, 2003. [KP10] S. Kuksin and G. Perelman. Vey theorem in infinite dimensions and its application to KdV. Discrete Contin. Dyn. Syst., 27(1):1–24, 2010. [KPV93] C. E. Kenig, G. Ponce, and L. Vega. Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle. Comm. Pure Appl. Math., 46(4):527–620, 1993. [KST12] T. Kappeler, B. Schaad, and P. Topalov. Asymptotics of spectral quantities of Schrödinger operators. In Spectral geometry, volume 84 of Proc. Sympos. Pure Math., pages 243–284. Amer. Math. Soc., Providence, RI, 2012. [KST13] T. Kappeler, B. Schaad, and P. Topalov. Qualitative features of periodic solutions of KdV. Comm. Partial Differential Equations, 38(9):1626–1673, 2013. [KT86] T. Kappeler and E. Trubowitz. Properties of the scattering map. Comment. Math. Helv., 61(3):442–480, 1986. [KT88] T. Kappeler and E. Trubowitz. Properties of the scattering map. II. Comment. Math. Helv., 63(1):150–167, 1988. [Kuk10] S. Kuksin. Damped-driven KdV and effective equations for long-time behaviour of its solutions. Geom. Funct. Anal., 20(6):1431–1463, 2010. [Mar86] V. Marchenko. Sturm-Liouville operators and applications. Birkhäuser, Basel, 1986.

129

[MBC14] A. Maiocchi, D. Bambusi, and A. Carati. An Averaging Theorem for FPU in the Thermodynamic Limit. J. Stat. Phys., 155(2):300–322, 2014. [McL75a] D. McLaughlin. Erratum: Four examples of the inverse method as a canonical transformation. Journal of Mathematical Physics, 16(8), 1975. [McL75b] D. McLaughlin. Four examples of the inverse method as a canonical transformation. J. Mathematical Phys., 16:96–99, 1975. [MS14] A. Maspero and B. Schaad. One smoothing properties of the KdV flow on R. In preparation, 2014. [Muj86] J. Mujica. Complex analysis in Banach spaces, volume 120 of North-Holland Mathematics Studies. North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions, Notas de Matemática [Mathematical Notes], 107. [Nik86] N. V. Nikolenko. The method of Poincaré normal forms in problems of integrability of equations of evolution type. Uspekhi Mat. Nauk, 41(5(251)):109–152, 263, 1986. [NMPZ84] S. Novikov, S. Manakov, L. Pitaevski˘ı, and V. Zakharov. Theory of solitons. Contemporary Soviet Mathematics. Consultants Bureau [Plenum], New York, 1984. The inverse scattering method, Translated from the Russian. [Nov96] R. G. Novikov. Inverse scattering up to smooth functions for the Schrödinger equation in dimension 1. Bull. Sci. Math., 120(5):473–491, 1996. [NP09] J. Nahas and G. Ponce. On the persistent properties of solutions to semi-linear Schrödinger equation. Comm. Partial Differential Equations, 34(10-12):1208–1227, 2009. [PCSF11] A. Ponno, H. Christodoulidi, Ch. Skokos, and S. Flach. The two-stage dynamics in the fermi-pasta-ulam problem: From regular to diffusive behavior. Chaos: An Interdisciplinary Journal of Nonlinear Science, 21(4):–, 2011. [SW00] G. Schneider and C. E. Wayne. Counter-propagating waves on fluid surfaces and the continuum limit of the Fermi-Pasta-Ulam model. In International Conference on Differential Equations, Vol. 1, 2 (Berlin, 1999), pages 390–404. World Sci. Publ., River Edge, NJ, 2000. [Tan74] S. Tanaka. Korteweg-de Vries equation: construction of solutions in terms of scattering data. Osaka J. Math., 11:49–59, 1974. [Tod67] M. Toda. Vibration of a Chain with Nonlinear Interaction. Journal of the Physical Society of Japan, 22:431, February 1967. [Trè70] F. Trèves. An abstract nonlinear Cauchy-Kovalevska theorem. Trans. Amer. Math. Soc., 150:77–92, 1970. [Vey78] J. Vey. Sur certains systèmes dynamiques séparables. Amer. J. Math., 100(3):591–614, 1978. 130

[vM76] P. van Moerbeke. The spectrum of Jacobi matrices. Invent. Math., 37(1):45–81, 1976. [ZF71] V. Zaharov and L. Faddeev. The Korteweg-de Vries equation is a fully integrable Hamiltonian system. Funkcional. Anal. i Priložen., 5(4):18–27, 1971. [Zho98] X. Zhou. L2 -Sobolev space bijectivity of the scattering and inverse scattering transforms. Comm. Pure Appl. Math., 51(7):697–731, 1998. [ZM74] V. Zaharov and S. Manakov. The complete integrability of the nonlinear Schrödinger equation. Teoret. Mat. Fiz., 19:332–343, 1974. [ZS71] V. Zakharov and A. Shabat. Exact theory of two-dimensional self-focusing and onedimensional self-modulation of waves in nonlinear media. Ž. Èksper. Teoret. Fiz., 61(1):118–134, 1971.

131